Download Algebraic Numbers and Number Fields

Sage Reference Manual: Algebraic
Numbers and Number Fields
Release 7.6
The Sage Development Team
Mar 25, 2017
CONTENTS
1
Algebraic Number Fields
2
Morphisms
179
3
Orders, Ideals, Ideal Classes
195
4
Totally Real Fields
267
5
Algebraic Numbers
279
6
Indices and Tables
343
Bibliography
1
345
i
ii
CHAPTER
ONE
ALGEBRAIC NUMBER FIELDS
1.1 Number Fields
AUTHORS:
• William Stein (2004, 2005): initial version
• Steven Sivek (2006-05-12): added support for relative extensions
• William Stein (2007-09-04): major rewrite and documentation
• Robert Bradshaw (2008-10): specified embeddings into ambient fields
• Simon King (2010-05): Improve coercion from GAP
• Jeroen Demeyer (2010-07, 2011-04): Upgrade PARI (trac ticket #9343, trac ticket #10430, trac ticket #11130)
• Robert Harron (2012-08): added is_CM(), complex_conjugation(), and maximal_totally_real_subfield()
• Christian Stump (2012-11): added conversion to universal cyclotomic field
• Julian Rueth (2014-04-03): absolute number fields are unique parents
• Vincent Delecroix (2015-02): comparisons/floor/ceil using embeddings
• Kiran Kedlaya (2016-05): relative number fields hash based on relative polynomials
• Peter Bruin (2016-06): make number fields fully satisfy unique representation
Note: Unlike in PARI/GP, class group computations in Sage do not by default assume the Generalized Riemann
Hypothesis. To do class groups computations not provably correctly you must often pass the flag proof=False to
functions or call the function proof.number_field(False) . It can easily take 1000’s of times longer to do
computations with proof=True (the default).
This example follows one in the Magma reference manual:
sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: z = y^5/11; z
420/11*y^3 - 40000/11*y
sage: R.<y> = PolynomialRing(K)
sage: f = y^2 + y + 1
sage: L.<a> = K.extension(f); L
Number Field in a with defining polynomial y^2 + y + 1 over its base field
sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field
We do some arithmetic in a tower of relative number fields:
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sage: K.<cuberoot2> = NumberField(x^3 - 2)
sage: L.<cuberoot3> = K.extension(x^3 - 3)
sage: S.<sqrt2> = L.extension(x^2 - 2)
sage: S
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: sqrt2 * cuberoot3
cuberoot3*sqrt2
sage: (sqrt2 + cuberoot3)^5
(20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
sage: cuberoot2 + cuberoot3
cuberoot3 + cuberoot2
sage: cuberoot2 + cuberoot3 + sqrt2
sqrt2 + cuberoot3 + cuberoot2
sage: (cuberoot2 + cuberoot3 + sqrt2)^2
(2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2
˓→+ 2
sage: cuberoot2 + sqrt2
sqrt2 + cuberoot2
sage: a = S(cuberoot2); a
cuberoot2
sage: a.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
Warning: Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY
SLOW. It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there
(which is quite fast).
class sage.rings.number_field.number_field. CyclotomicFieldFactory
Bases: sage.structure.factory.UniqueFactory
Return the 𝑛-th cyclotomic field, where n is a positive integer, or the universal cyclotomic field if n==0 .
For the documentation of the universal cyclotomic field, see UniversalCyclotomicField .
INPUT:
•n - a nonnegative integer, default:0
•names - name of generator (optional - defaults to zetan)
•bracket - Defines the brackets in the case of n==0 , and is ignored otherwise. Can be any even length
string, with "()" being the default.
•embedding - bool or n-th root of unity in an ambient field (default True)
EXAMPLES:
If called without a parameter, we get the universal cyclotomic field :
sage: CyclotomicField()
Universal Cyclotomic Field
We create the 7th cyclotomic field Q(𝜁7 ) with the default generator name.
sage: k = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
zeta7
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The default embedding sends the generator to the complex primitive 𝑛𝑡ℎ root of unity of least argument.
sage: CC(k.gen())
0.623489801858734 + 0.781831482468030*I
Cyclotomic fields are of a special type.
sage: type(k)
<class 'sage.rings.number_field.number_field.NumberField_cyclotomic_with_category
˓→'>
We can specify a different generator name as follows.
sage: k.<z7> = CyclotomicField(7); k
Cyclotomic Field of order 7 and degree 6
sage: k.gen()
z7
The 𝑛 must be an integer.
sage: CyclotomicField(3/2)
Traceback (most recent call last):
...
TypeError: no conversion of this rational to integer
The degree must be nonnegative.
sage: CyclotomicField(-1)
Traceback (most recent call last):
...
ValueError: n (=-1) must be a positive integer
The special case 𝑛 = 1 does not return the rational numbers:
sage: CyclotomicField(1)
Cyclotomic Field of order 1 and degree 1
Due to their default embedding into C, cyclotomic number fields are all compatible.
sage: cf30 = CyclotomicField(30)
sage: cf5 = CyclotomicField(5)
sage: cf3 = CyclotomicField(3)
sage: cf30.gen() + cf5.gen() + cf3.gen()
zeta30^6 + zeta30^5 + zeta30 - 1
sage: cf6 = CyclotomicField(6) ; z6 = cf6.0
sage: cf3 = CyclotomicField(3) ; z3 = cf3.0
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: cf9 = CyclotomicField(9) ; z9 = cf9.0
sage: cf18 = CyclotomicField(18) ; z18 = cf18.0
sage: cf18(z9)
zeta18^2
sage: cf9(z18)
-zeta9^5
sage: cf18(z3)
zeta18^3 - 1
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sage: cf18(z6)
zeta18^3
sage: cf18(z6)**2
zeta18^3 - 1
sage: cf9(z3)
zeta9^3
create_key ( n=0, names=None, embedding=True)
Create the unique key for the cyclotomic field specified by the parameters.
create_object ( version, key, **extra_args)
Create the unique cyclotomic field defined by key .
sage.rings.number_field.number_field. NumberField ( polynomial,
name=None,
check=True, names=None, embedding=None,
latex_name=None,
assume_disc_small=False, maximize_at_primes=None,
structure=None)
Return the number field (or tower of number fields) defined by the irreducible polynomial .
INPUT:
•polynomial - a polynomial over Q or a number field, or a list of such polynomials.
•name - a string or a list of strings, the names of the generators
•check - a boolean (default: True ); do type checking and irreducibility checking.
•embedding - None , an element, or a list of elements, the images of the generators in an ambient field
(default: None )
•latex_name - None , a string, or a list of strings (default: None ), how the generators are printed for
latex output
•assume_disc_small – a boolean (default: False ); if True , assume that no square of a prime
greater than PARI’s primelimit (which should be 500000); only applies for absolute fields at present.
•maximize_at_primes – None or a list of primes (default: None ); if not None , then the maximal
order is computed by maximizing only at the primes in this list, which completely avoids having to factor
the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
•structure – None , a list or an instance of structure.NumberFieldStructure (default:
None ), internally used to pass in additional structural information, e.g., about the field from which this
field is created as a subfield.
EXAMPLES:
sage: z = QQ['z'].0
sage: K = NumberField(z^2 - 2,'s'); K
Number Field in s with defining polynomial z^2 - 2
sage: s = K.0; s
s
sage: s*s
2
sage: s^2
2
Constructing a relative number field:
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sage: K.<a> = NumberField(x^2 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^3+t+a); L
Number Field in b with defining polynomial t^3 + t + a over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial x^6 + 2*x^4 + x^2 - 2
sage: a*b
a*b
sage: L(a)
a
sage: L.lift_to_base(b^3 + b)
-a
Constructing another number field:
sage: k.<i> = NumberField(x^2 + 1)
sage: R.<z> = k[]
sage: m.<j> = NumberField(z^3 + i*z + 3)
sage: m
Number Field in j with defining polynomial z^3 + i*z + 3 over its base field
Number fields are globally unique:
sage:
sage:
5
sage:
sage:
True
K.<a> = NumberField(x^3 - 5)
a^3
L.<a> = NumberField(x^3 - 5)
K is L
Equality of number fields depends on the variable name of the defining polynomial:
sage: x = polygen(QQ, 'x'); y =
sage: k.<a> = NumberField(x^2 +
sage: m.<a> = NumberField(y^2 +
sage: k
Number Field in a with defining
sage: m
Number Field in a with defining
sage: k == m
False
polygen(QQ, 'y')
3)
3)
polynomial x^2 + 3
polynomial y^2 + 3
In case of conflict of the generator name with the name given by the preparser, the name given by the preparser
takes precedence:
sage: K.<b> = NumberField(x^2 + 5, 'a'); K
Number Field in b with defining polynomial x^2 + 5
One can also define number fields with specified embeddings, may be used for arithmetic and deduce relations
with other number fields which would not be valid for an abstract number field.
sage: K.<a> = NumberField(x^3-2, embedding=1.2)
sage: RR.coerce_map_from(K)
Composite map:
From: Number Field in a with defining polynomial x^3 - 2
To:
Real Field with 53 bits of precision
Defn:
Generic morphism:
From: Number Field in a with defining polynomial x^3 - 2
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To:
Real Lazy Field
Defn: a -> 1.259921049894873?
then
Conversion via _mpfr_ method map:
From: Real Lazy Field
To:
Real Field with 53 bits of precision
sage: RR(a)
1.25992104989487
sage: 1.1 + a
2.35992104989487
sage: b = 1/(a+1); b
1/3*a^2 - 1/3*a + 1/3
sage: RR(b)
0.442493334024442
sage: L.<b> = NumberField(x^6-2, embedding=1.1)
sage: L(a)
b^2
sage: a + b
b^2 + b
Note that the image only needs to be specified to enough precision to distinguish roots, and is exactly computed
to any needed precision:
sage: RealField(200)(a)
1.2599210498948731647672106072782283505702514647015079800820
One can embed into any other field:
sage: K.<a> = NumberField(x^3-2, embedding=CC.gen()-0.6)
sage: CC(a)
-0.629960524947436 + 1.09112363597172*I
sage: L = Qp(5)
sage: f = polygen(L)^3 - 2
sage: K.<a> = NumberField(x^3-2, embedding=f.roots()[0][0])
sage: a + L(1)
4 + 2*5^2 + 2*5^3 + 3*5^4 + 5^5 + 4*5^6 + 2*5^8 + 3*5^9 + 4*5^12 + 4*5^14 + 4*5^
˓→15 + 3*5^16 + 5^17 + 5^18 + 2*5^19 + O(5^20)
sage: L.<b> = NumberField(x^6-x^2+1/10, embedding=1)
sage: K.<a> = NumberField(x^3-x+1/10, embedding=b^2)
sage: a+b
b^2 + b
sage: CC(a) == CC(b)^2
True
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 1/10
To:
Number Field in b with defining polynomial x^6 - x^2 + 1/10
Defn: a -> b^2
The QuadraticField and CyclotomicField constructors create an embedding by default unless otherwise specified:
sage: K.<zeta> = CyclotomicField(15)
sage: CC(zeta)
0.913545457642601 + 0.406736643075800*I
sage: L.<sqrtn3> = QuadraticField(-3)
sage: K(sqrtn3)
2*zeta^5 + 1
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sage: sqrtn3 + zeta
2*zeta^5 + zeta + 1
Comparison depends on the (real) embedding specified (or the one selected by default). Note that the codomain
of the embedding must be 𝑄𝑄𝑏𝑎𝑟 or 𝐴𝐴 for this to work (see trac ticket #20184):
sage: N.<g> = NumberField(x^3+2,embedding=1)
sage: 1 < g
False
sage: g > 1
False
sage: RR(g)
-1.25992104989487
If no embedding is specified or is complex, the comparison is not returning something meaningful.:
sage: N.<g> = NumberField(x^3+2)
sage: 1 < g
False
sage: g > 1
True
Since SageMath 6.9, number fields may be defined by polynomials that are not necessarily integral or monic.
The only notable practical point is that in the PARI interface, a monic integral polynomial defining the same
number field is computed and used:
sage: K.<a> = NumberField(2*x^3 + x + 1)
sage: K.pari_polynomial()
x^3 - x^2 - 2
Elements and ideals may be converted to and from PARI as follows:
sage: pari(a)
Mod(-1/2*y^2 + 1/2*y, y^3 - y^2 - 2)
sage: K(pari(a))
a
sage: I = K.ideal(a); I
Fractional ideal (a)
sage: I.pari_hnf()
[1, 0, 0; 0, 1, 0; 0, 0, 1/2]
sage: K.ideal(I.pari_hnf())
Fractional ideal (a)
Here is an example where the field has non-trivial class group:
sage: L.<b> = NumberField(3*x^2 - 1/5)
sage: L.pari_polynomial()
x^2 - 15
sage: J = L.primes_above(2)[0]; J
Fractional ideal (2, 15*b + 1)
sage: J.pari_hnf()
[2, 1; 0, 1]
sage: L.ideal(J.pari_hnf())
Fractional ideal (2, 15*b + 1)
An example involving a variable name that defines a function in PARI:
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sage: theta = polygen(QQ, 'theta')
sage: M.<z> = NumberField([theta^3 + 4, theta^2 + 3]); M
Number Field in z0 with defining polynomial theta^3 + 4 over its base field
sage: L = NumberField(y^3 + y + 3, 'a'); L
Number Field in a with defining polynomial y^3 + y + 3
sage: L.defining_polynomial().parent()
Univariate Polynomial Ring in y over Rational Field
sage: W1 = NumberField(x^2+1,'a')
sage: K.<x> = CyclotomicField(5)[]
sage: W.<a> = NumberField(x^2 + 1); W
Number Field in a with defining polynomial x^2 + 1 over its base field
The following has been fixed in trac ticket #8800:
sage:
sage:
sage:
sage:
sage:
True
P.<x> = QQ[]
K.<a> = NumberField(x^3-5,embedding=0)
L.<b> = K.extension(x^2+a)
F, R = L.construction()
F(R) == L
# indirect doctest
Check that trac ticket #11670 has been fixed:
sage:
sage:
True
sage:
sage:
True
sage:
sage:
True
K.<a> = NumberField(x^2 - x - 1)
loads(dumps(K)) is K
K.<a> = NumberField(x^3 - x - 1)
loads(dumps(K)) is K
K.<a> = CyclotomicField(7)
loads(dumps(K)) is K
Another problem that was found while working on trac ticket #11670, maximize_at_primes
assume_disc_small were lost when pickling:
and
sage: K.<a> = NumberField(x^3-2, assume_disc_small=True, maximize_at_primes=[2],
˓→latex_name='\\alpha', embedding=2^(1/3))
sage: L = loads(dumps(K))
sage: L._assume_disc_small
True
sage: L._maximize_at_primes
(2,)
It is an error not to specify the generator:
sage: K = NumberField(x^2-2)
Traceback (most recent call last):
...
TypeError: You must specify the name of the generator.
class sage.rings.number_field.number_field. NumberFieldFactory
Bases: sage.structure.factory.UniqueFactory
Factory for number fields.
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This should usually not be called directly, use NumberField() instead.
INPUT:
•polynomial - a polynomial over Q or a number field.
•name - a string (default: 'a' ), the name of the generator
•check - a boolean (default: True ); do type checking and irreducibility checking.
•embedding - None or an element, the images of the generator in an ambient field (default: None )
•latex_name - None or a string (default: None ), how the generator is printed for latex output
•assume_disc_small – a boolean (default: False ); if True , assume that no square of a prime
greater than PARI’s primelimit (which should be 500000); only applies for absolute fields at present.
•maximize_at_primes – None or a list of primes (default: None ); if not None , then the maximal
order is computed by maximizing only at the primes in this list, which completely avoids having to factor
the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
•structure – None or an instance of structure.NumberFieldStructure (default: None ),
internally used to pass in additional structural information, e.g., about the field from which this field is
created as a subfield.
create_key_and_extra_args ( polynomial, name, check, embedding, latex_name,
sume_disc_small, maximize_at_primes, structure)
Create a unique key for the number field specified by the parameters.
as-
create_object ( version, key, check)
Create the unique number field defined by key .
sage.rings.number_field.number_field. NumberFieldTower ( polynomials,
names,
check=True,
embeddings=None,
latex_names=None,
assume_disc_small=False,
maximize_at_primes=None,
structures=None)
Create the tower of number fields defined by the polynomials in the list polynomials .
INPUT:
•polynomials - a list of polynomials. Each entry must be polynomial which is irreducible over the
number field generated by the roots of the following entries.
•names - a list of strings or a string, the names of the generators of the relative number fields. If a single
string, then names are generated from that string.
•check - a boolean (default: True ), whether to check that the polynomials are irreducible
•embeddings - a list of elemenst or None (default: None ), embeddings of the relative number fields
in an ambient field.
•latex_names - a list of strings or None (default: None ), names used to print the generators for latex
output.
•assume_disc_small – a boolean (default: False ); if True , assume that no square of a prime
greater than PARI’s primelimit (which should be 500000); only applies for absolute fields at present.
•maximize_at_primes – None or a list of primes (default: None ); if not None , then the maximal
order is computed by maximizing only at the primes in this list, which completely avoids having to factor
the discriminant, but of course can lead to wrong results; only applies for absolute fields at present.
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•structures – None or a list (default: None ), internally used to provide additional information about
the number field such as the field from which it was created.
OUTPUT:
Returns the relative number field generated by a root of the first entry of polynomials over the relative
number field generated by root of the second entry of polynomials ... over the number field over which the
last entry of polynomials is defined.
EXAMPLES:
sage: k.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5]); k # indirect doctest
Number Field in a with defining polynomial x^2 + 1 over its base field
sage: a^2
-1
sage: b^2
-3
sage: c^2
-5
sage: (a+b+c)^2
(2*b + 2*c)*a + 2*c*b - 9
The Galois group is a product of 3 groups of order 2:
sage: k.galois_group(type="pari")
Galois group PARI group [8, 1, 3, "E(8)=2[x]2[x]2"] of degree 8 of the Number
˓→Field in a with defining polynomial x^2 + 1 over its base field
Repeatedly calling base_field allows us to descend the internally constructed tower of fields:
sage: k.base_field()
Number Field in b with defining polynomial x^2 + 3 over its base field
sage: k.base_field().base_field()
Number Field in c with defining polynomial x^2 + 5
sage: k.base_field().base_field().base_field()
Rational Field
In the following example the second polynomial is reducible over the first, so we get an error:
sage: v = NumberField([x^3 - 2, x^3 - 2], names='a')
Traceback (most recent call last):
...
ValueError: defining polynomial (x^3 - 2) must be irreducible
We mix polynomial parent rings:
sage: k.<y> = QQ[]
sage: m = NumberField([y^3 - 3, x^2 + x + 1, y^3 + 2], 'beta')
sage: m
Number Field in beta0 with defining polynomial y^3 - 3 over its base field
sage: m.base_field ()
Number Field in beta1 with defining polynomial x^2 + x + 1 over its base field
A tower of quadratic fields:
sage: K.<a> = NumberField([x^2 + 3, x^2 + 2, x^2 + 1])
sage: K
Number Field in a0 with defining polynomial x^2 + 3 over its base field
sage: K.base_field()
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Number Field in a1 with defining polynomial x^2 + 2 over its base field
sage: K.base_field().base_field()
Number Field in a2 with defining polynomial x^2 + 1
A bigger tower of quadratic fields:
sage: K.<a2,a3,a5,a7> = NumberField([x^2 + p for p in [2,3,5,7]]); K
Number Field in a2 with defining polynomial x^2 + 2 over its base field
sage: a2^2
-2
sage: a3^2
-3
sage: (a2+a3+a5+a7)^3
((6*a5 + 6*a7)*a3 + 6*a7*a5 - 47)*a2 + (6*a7*a5 - 45)*a3 - 41*a5 - 37*a7
The function can also be called by name:
sage: NumberFieldTower([x^2 + 1, x^2 + 2], ['a','b'])
Number Field in a with defining polynomial x^2 + 1 over its base field
class sage.rings.number_field.number_field. NumberField_absolute ( polynomial,
name,
latex_name=None,
check=True,
embedding=None, assume_disc_small=False,
maximize_at_primes=None,
structure=None)
Bases: sage.rings.number_field.number_field.NumberField_generic
Function to initialize an absolute number field.
EXAMPLES:
sage: K = NumberField(x^17 + 3, 'a'); K
Number Field in a with defining polynomial x^17 + 3
sage: type(K)
<class 'sage.rings.number_field.number_field.NumberField_absolute_with_category'>
sage: TestSuite(K).run()
Minkowski_embedding ( B=None, prec=None)
Return an nxn matrix over RDF whose columns are the images of the basis {1, 𝛼, . . . , 𝛼𝑛−1 } of self over
Q (as vector spaces), where here 𝛼 is the generator of self over Q, i.e. self.gen(0). If B is not None, return
the images of the vectors in B as the columns instead. If prec is not None, use RealField(prec) instead of
RDF.
This embedding is the so-called “Minkowski embedding” of a number field in R𝑛 : given the 𝑛 embeddings
𝜎1 , . . . , 𝜎𝑛 of self in C, write 𝜎1 , . . . , 𝜎𝑟 for the real embeddings, and 𝜎𝑟+1 , . . . , 𝜎𝑟+𝑠 for choices of one
of each pair of complex conjugate embeddings (in our case, we simply choose the one where the image of
𝛼 has positive real part). Here (𝑟, 𝑠) is the signature of self. Then the Minkowski embedding is given by
√
√
√
√
𝑥 ↦→ (𝜎1 (𝑥), . . . , 𝜎𝑟 (𝑥), 2ℜ(𝜎𝑟+1 (𝑥)), 2ℑ(𝜎𝑟+1 (𝑥)), . . . , 2ℜ(𝜎𝑟+𝑠 (𝑥)), 2ℑ(𝜎𝑟+𝑠 (𝑥)))
Equivalently,
this is an embedding of self in R𝑛 so that the usual norm on R𝑛 coincides with |𝑥| =
∑︀
2
𝑖 |𝜎𝑖 (𝑥)| on self.
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TODO: This could be much improved by implementing homomorphisms over VectorSpaces.
EXAMPLES:
sage: F.<alpha> = NumberField(x^3+2)
sage: F.Minkowski_embedding()
[ 1.00000000000000 -1.25992104989487 1.58740105196820]
[ 1.41421356237... 0.8908987181... -1.12246204830...]
[0.000000000000000 1.54308184421... 1.94416129723...]
sage: F.Minkowski_embedding([1, alpha+2, alpha^2-alpha])
[ 1.00000000000000 0.740078950105127 2.84732210186307]
[ 1.41421356237... 3.7193258428... -2.01336076644...]
[0.000000000000000 1.54308184421... 0.40107945302...]
sage: F.Minkowski_embedding() * (alpha + 2).vector().column()
[0.740078950105127]
[ 3.7193258428...]
[ 1.54308184421...]
absolute_degree ( )
A synonym for degree.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_degree()
2
absolute_different ( )
A synonym for different.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_different()
Fractional ideal (2)
absolute_discriminant ( )
A synonym for discriminant.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_discriminant()
-4
absolute_generator ( )
An alias for sage.rings.number_field.number_field.NumberField_generic.gen()
.
This is provided for consistency with relative fields, where the element returned by
sage.rings.number_field.number_field_rel.NumberField_relative.gen()
only generates the field over its base field (not necessarily over Q).
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 17)
sage: K.absolute_generator()
a
absolute_polynomial ( )
A synonym for polynomial.
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EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_polynomial()
x^2 + 1
absolute_vector_space ( )
A synonym for vector_space.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.absolute_vector_space()
(Vector space of dimension 2 over Rational Field,
Isomorphism map:
From: Vector space of dimension 2 over Rational Field
To:
Number Field in i with defining polynomial x^2 + 1,
Isomorphism map:
From: Number Field in i with defining polynomial x^2 + 1
To:
Vector space of dimension 2 over Rational Field)
automorphisms ( )
Compute all Galois automorphisms of self.
This uses PARI’s pari:nfgaloisconj and is much faster than root finding for many fields.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 10000)
sage: K.automorphisms()
[
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial x^2 + 10000
Defn: a |--> -a
]
Here’s a larger example, that would take some time if we found roots instead of using PARI’s specialized
machinery:
sage: K = NumberField(x^6 - x^4 - 2*x^2 + 1, 'a')
sage: len(K.automorphisms())
2
𝐿 is the Galois closure of 𝐾:
sage: L = NumberField(x^24 - 84*x^22 + 2814*x^20 - 15880*x^18 - 409563*x^16 ˓→8543892*x^14 + 25518202*x^12 + 32831026956*x^10 - 672691027218*x^8 ˓→4985379093428*x^6 + 320854419319140*x^4 + 817662865724712*x^2 +
˓→513191437605441, 'a')
sage: len(L.automorphisms())
24
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage:
sage:
sage:
sage:
R.<x> = QQ[]
f = 7/9*x^3 + 7/3*x^2 - 56*x + 123
K.<a> = NumberField(f)
A = K.automorphisms(); A
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[
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/
˓→3*x^2 - 56*x + 123
Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/
˓→3*x^2 - 56*x + 123
Defn: a |--> -7/15*a^2 - 18/5*a + 96/5,
Ring endomorphism of Number Field in a with defining polynomial 7/9*x^3 + 7/
˓→3*x^2 - 56*x + 123
Defn: a |--> 7/15*a^2 + 13/5*a - 111/5
]
sage: prod(x - sigma(a) for sigma in A) == f.monic()
True
base_field ( )
Returns the base field of self, which is always QQ
EXAMPLES:
sage: K = CyclotomicField(5)
sage: K.base_field()
Rational Field
change_names ( names)
Return number field isomorphic to self but with the given generator name.
INPUT:
•names - should be exactly one variable name.
Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and
to_K is an isomorphism from self to K.
EXAMPLES:
sage: K.<z> = NumberField(x^2 + 3); K
Number Field in z with defining polynomial x^2 + 3
sage: L.<ww> = K.change_names()
sage: L
Number Field in ww with defining polynomial x^2 + 3
sage: L.structure()[0]
Isomorphism given by variable name change map:
From: Number Field in ww with defining polynomial x^2 + 3
To:
Number Field in z with defining polynomial x^2 + 3
sage: L.structure()[0](ww + 5/3)
z + 5/3
elements_of_bounded_height ( bound, precision=53, LLL=False)
Return an iterator over the elements of self with relative multiplicative height at most bound .
The algorithm requires floating point arithmetic, so the user is allowed to specify the precision for such
calculations.
It might be helpful to work with an LLL-reduced system of fundamental units, so the user has the option
to perform an LLL reduction for the fundamental units by setting LLL to True.
Certain computations may be faster assuming GRH, which may be done globally by using the number_field(True/False) switch.
For details: See [Doyle-Krumm].
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INPUT:
•bound - a real number
•precision - (default: 53) a positive integer
•LLL - (default: False) a boolean value
OUTPUT:
•an iterator of number field elements
Warning: In the current implementation, the output of the algorithm cannot be guaranteed to be
correct due to the necessity of floating point computations. In some cases, the default 53-bit precision
is considerably lower than would be required for the algorithm to generate correct output.
Todo
Should implement a version of the algorithm that guarantees correct output. See Algorithm 4 in [DoyleKrumm] for details of an implementation that takes precision issues into account.
EXAMPLES:
There are no elements in a number field with multiplicative height less than 1:
sage: K.<g> = NumberField(x^5 - x + 19)
sage: list(K.elements_of_bounded_height(0.9))
[]
The only elements in a number field of height 1 are 0 and the roots of unity:
sage: K.<a> = NumberField(x^2 + x + 1)
sage: list(K.elements_of_bounded_height(1))
[0, a + 1, a, -1, -a - 1, -a, 1]
sage: K.<a> = CyclotomicField(20)
sage: len(list(K.elements_of_bounded_height(1)))
21
The elements in the output iterator all have relative multiplicative height at most the input bound:
sage: K.<a> = NumberField(x^6 + 2)
sage: L = K.elements_of_bounded_height(5)
sage: for t in L:
....:
exp(6*t.global_height())
....:
1.00000000000000
1.00000000000000
1.00000000000000
2.00000000000000
2.00000000000000
2.00000000000000
2.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000
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sage: K.<a> = NumberField(x^2 - 71)
sage: L = K.elements_of_bounded_height(20)
sage: all(exp(2*t.global_height()) <= 20 for t in L) # long time (5 s)
True
sage: K.<a> = NumberField(x^2 + 17)
sage: L = K.elements_of_bounded_height(120)
sage: len(list(L))
9047
sage: K.<a> = NumberField(x^4 - 5)
sage: L = K.elements_of_bounded_height(50)
sage: len(list(L)) # long time (2 s)
2163
sage: K.<a> = CyclotomicField(13)
sage: L = K.elements_of_bounded_height(2)
sage: len(list(L)) # long time (3 s)
27
sage: K.<a> = NumberField(x^6 + 2)
sage: L = K.elements_of_bounded_height(60, precision=100)
sage: len(list(L)) # long time (5 s)
1899
sage: K.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
sage: L = K.elements_of_bounded_height(10, LLL=true)
sage: len(list(L))
99
AUTHORS:
•John Doyle (2013)
•David Krumm (2013)
embeddings ( K)
Compute all field embeddings of self into the field K (which need not even be a number field, e.g., it could
be the complex numbers). This will return an identical result when given K as input again.
If possible, the most natural embedding of self into K is put first in the list.
INPUT:
•K - a number field
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2)
sage: L.<a1> = K.galois_closure(); L
Number Field in a1 with defining polynomial x^6 + 108
sage: K.embeddings(L)[0]
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Number Field in a1 with defining polynomial x^6 + 108
Defn: a |--> 1/18*a1^4
sage: K.embeddings(L) is K.embeddings(L)
True
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We embed a quadratic field into a cyclotomic field:
sage: L.<a> = QuadraticField(-7)
sage: K = CyclotomicField(7)
sage: L.embeddings(K)
[
Ring morphism:
From: Number Field in a with defining polynomial x^2 + 7
To:
Cyclotomic Field of order 7 and degree 6
Defn: a |--> 2*zeta7^4 + 2*zeta7^2 + 2*zeta7 + 1,
Ring morphism:
From: Number Field in a with defining polynomial x^2 + 7
To:
Cyclotomic Field of order 7 and degree 6
Defn: a |--> -2*zeta7^4 - 2*zeta7^2 - 2*zeta7 - 1
]
We embed a cubic field in the complex numbers:
sage: K.<a> = NumberField(x^3 - 2)
sage: K.embeddings(CC)
[
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... - 1.09112363597172*I,
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... + 1.09112363597172*I,
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Complex Field with 53 bits of precision
Defn: a |--> 1.25992104989487
]
Test that trac ticket #15053 is fixed:
sage: K = NumberField(x^3 - 2, 'a')
sage: K.embeddings(GF(3))
[]
galois_closure ( names=None, map=False)
Return number field 𝐾 that is the Galois closure of self, i.e., is generated by all roots of the defining
polynomial of self, and possibly an embedding of self into 𝐾.
INPUT:
•names - variable name for Galois closure
•map - (default: False) also return an embedding of self into 𝐾
EXAMPLES:
sage: K.<a> = NumberField(x^4 - 2)
sage: M = K.galois_closure('b'); M
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: L.<a2> = K.galois_closure(); L
Number Field in a2 with defining polynomial x^8 + 28*x^4 + 2500
sage: K.galois_group(names=("a3")).order()
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sage: phi = K.embeddings(L)[0]
sage: phi(K.0)
1/120*a2^5 + 19/60*a2
sage: phi(K.0).minpoly()
x^4 - 2
sage: L, phi = K.galois_closure('b', map=True)
sage: L
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
sage: phi
Ring morphism:
From: Number Field in a with defining polynomial x^4 - 2
To:
Number Field in b with defining polynomial x^8 + 28*x^4 + 2500
Defn: a |--> 1/240*b^5 - 41/120*b
A cyclotomic field is already Galois:
sage: K.<a> = NumberField(cyclotomic_polynomial(23))
sage: L.<z> = K.galois_closure()
sage: L
Number Field in z with defining polynomial x^22 + x^21 + x^20 + x^19 + x^18 +
˓→x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^
˓→6 + x^5 + x^4 + x^3 + x^2 + x + 1
hilbert_conductor ( a, b)
This is the product of all (finite) primes where the Hilbert symbol is -1. What is the same, this is the
(reduced) discriminant of the quaternion algebra (𝑎, 𝑏) over a number field.
INPUT:
•a , b – elements of the number field self
OUTPUT:
•squarefree ideal of the ring of integers of self
EXAMPLES:
sage: F.<a> = NumberField(x^2-x-1)
sage: F.hilbert_conductor(2*a,F(-1))
Fractional ideal (2)
sage: K.<b> = NumberField(x^3-4*x+2)
sage: K.hilbert_conductor(K(2),K(-2))
Fractional ideal (1)
sage: K.hilbert_conductor(K(2*b),K(-2))
Fractional ideal (b^2 + b - 2)
AUTHOR:
•Aly Deines
hilbert_symbol ( a, b, P=None)
Returns the Hilbert symbol (𝑎, 𝑏)𝑃 for a prime P of self and non-zero elements a and b of self. If P is
omitted, return the global Hilbert symbol (𝑎, 𝑏) instead.
INPUT:
•a , b – elements of self
•P – (default: None) If 𝑃 is None , compute the global symbol. Otherwise, 𝑃 should be either a
prime ideal of self (which may also be given as a generator or set of generators) or a real or complex
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embedding.
OUTPUT:
If a or b is zero, returns 0.
If a and b are non-zero and P is specified, returns the Hilbert symbol (𝑎, 𝑏)𝑃 , which is 1 if the equation
𝑎𝑥2 + 𝑏𝑦 2 = 1 has a solution in the completion of self at P, and is -1 otherwise.
If a and b are non-zero and P is unspecified, returns 1 if the equation has a solution in self and -1 otherwise.
EXAMPLES:
Some global examples:
sage:
sage:
0
sage:
0
sage:
1
sage:
-1
sage:
-1
K.<a> = NumberField(x^2 - 23)
K.hilbert_symbol(0, a+5)
K.hilbert_symbol(a, 0)
K.hilbert_symbol(-a, a+1)
K.hilbert_symbol(-a, a+2)
K.hilbert_symbol(a, a+5)
That the latter two are unsolvable should be visible in local obstructions. For the first, this is a prime ideal
above 19. For the second, the ramified prime above 23:
sage: K.hilbert_symbol(-a, a+2, a+2)
-1
sage: K.hilbert_symbol(a, a+5, K.ideal(23).factor()[0][0])
-1
More local examples:
sage:
0
sage:
1
sage:
-1
sage:
sage:
-1
sage:
1
K.hilbert_symbol(a, 0, K.ideal(5))
K.hilbert_symbol(a, a+5, K.ideal(5))
K.hilbert_symbol(a+1, 13, (a+6)*K.maximal_order())
[emb1, emb2] = K.embeddings(AA)
K.hilbert_symbol(a, -1, emb1)
K.hilbert_symbol(a, -1, emb2)
Ideals P can be given by generators:
sage:
sage:
sage:
1
sage:
sage:
1
K.<a> = NumberField(x^5 - 23)
pi = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11
K.hilbert_symbol(a, a+5, pi)
rho = 2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11
K.hilbert_symbol(a, a+5, rho)
This also works for non-principal ideals:
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sage:
sage:
sage:
(3, a
sage:
1
sage:
1
K.<a> = QuadraticField(-5)
P = K.ideal(3).factor()[0][0]
P.gens_reduced() # random, could be the other factor
+ 1)
K.hilbert_symbol(a, a+3, P)
K.hilbert_symbol(a, a+3, [3, a+1])
Primes above 2:
sage:
sage:
sage:
sage:
1
sage:
1
sage:
-1
K.<a> = NumberField(x^5 - 23)
O = K.maximal_order()
p = [p[0] for p in (2*O).factor() if p[0].norm() == 16][0]
K.hilbert_symbol(a, a+5, p)
K.hilbert_symbol(a, 2, p)
K.hilbert_symbol(-1, a-2, p)
Various real fields are allowed:
sage:
sage:
1
sage:
-1
sage:
[-1]
K.<a> = NumberField(x^3+x+1)
K.hilbert_symbol(a/3, 1/2, K.embeddings(RDF)[0])
K.hilbert_symbol(a/5, -1, K.embeddings(RR)[0])
[K.hilbert_symbol(a, -1, e) for e in K.embeddings(AA)]
Real embeddings are not allowed to be disguised as complex embeddings:
sage: K.<a> = QuadraticField(5)
sage: K.hilbert_symbol(-1, -1, K.embeddings(CC)[0])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
From: Number Field in a with defining polynomial x^2 - 5
To:
Complex Field with 53 bits of precision
Defn: a |--> -2.23606797749979) given as complex embedding in hilbert_
˓→symbol. Is it real or complex?
sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])
Traceback (most recent call last):
...
ValueError: Possibly real place (=Ring morphism:
From: Number Field in a with defining polynomial x^2 - 5
To:
Algebraic Field
Defn: a |--> -2.236067977499790?) given as complex embedding in hilbert_
˓→symbol. Is it real or complex?
sage: K.<b> = QuadraticField(-5)
sage: K.hilbert_symbol(-1, -1, K.embeddings(CDF)[0])
1
sage: K.hilbert_symbol(-1, -1, K.embeddings(QQbar)[0])
1
a and b do not have to be integral or coprime:
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sage:
sage:
sage:
1
sage:
sage:
1
sage:
-1
sage:
-1
sage:
1
sage:
sage:
1
K.<i> = QuadraticField(-1)
O = K.maximal_order()
K.hilbert_symbol(1/2, 1/6, 3*O)
p = 1+i
K.hilbert_symbol(p, p, p)
K.hilbert_symbol(p, 3*p, p)
K.hilbert_symbol(3, p, p)
K.hilbert_symbol(1/3, 1/5, 1+i)
L = QuadraticField(5, 'a')
L.hilbert_symbol(-3, -1/2, 2)
Various other examples:
sage:
sage:
1
sage:
sage:
sage:
sage:
sage:
1
sage:
1
sage:
sage:
sage:
1
sage:
1
sage:
-1
sage:
sage:
1
K.<a> = NumberField(x^3+x+1)
K.hilbert_symbol(-6912, 24, -a^2-a-2)
K.<a> = NumberField(x^5-23)
P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)
Q = K.ideal(-7*a^4 + 13*a^3 - 13*a^2 - 2*a + 50)
b = -a+5
K.hilbert_symbol(a,b,P)
K.hilbert_symbol(a,b,Q)
K.<a> = NumberField(x^5-23)
P = K.ideal(-1105*a^4 + 1541*a^3 - 795*a^2 - 2993*a + 11853)
K.hilbert_symbol(a, a+5, P)
K.hilbert_symbol(a, 2, P)
K.hilbert_symbol(a+5, 2, P)
K.<a> = NumberField(x^3 - 4*x + 2)
K.hilbert_symbol(2, -2, K.primes_above(2)[0])
Check that the bug reported at trac ticket #16043 has been fixed:
sage: K.<a> = NumberField(x^2 + 5)
sage: p = K.primes_above(2)[0]; p
Fractional ideal (2, a + 1)
sage: K.hilbert_symbol(2*a, -1, p)
1
sage: K.hilbert_symbol(2*a, 2, p)
-1
sage: K.hilbert_symbol(2*a, -2, p)
-1
AUTHOR:
•Aly Deines (2010-08-19): part of the doctests
•Marco Streng (2010-12-06)
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is_absolute ( )
Returns True since self is an absolute field.
EXAMPLES:
sage: K = CyclotomicField(5)
sage: K.is_absolute()
True
maximal_order ( v=None)
Return the maximal order, i.e., the ring of integers, associated to this number field.
INPUT:
•v - (default: None ) None , a prime, or a list of primes.
–if v is None , return the maximal order.
–if v is a prime, return an order that is 𝑝-maximal.
–if v is a list, return an order that is maximal at each prime in the list v .
EXAMPLES:
In this example, the maximal order cannot be generated by a single element:
sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)
sage: o = k.maximal_order()
sage: o
Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x +
˓→8
We compute 𝑝-maximal orders for several 𝑝. Note that computing a 𝑝-maximal order is much faster in
general than computing the maximal order:
sage: p = next_prime(10^22); q = next_prime(10^23)
sage: K.<a> = NumberField(x^3 - p*q)
sage: K.maximal_order([3]).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]
sage: K.maximal_order([2]).basis()
[1, a, a^2]
sage: K.maximal_order([p]).basis()
[1, a, a^2]
sage: K.maximal_order([q]).basis()
[1, a, a^2]
sage: K.maximal_order([p,3]).basis()
[1/3*a^2 + 1/3*a + 1/3, a, a^2]
An example with bigger discriminant:
sage: p = next_prime(10^97); q = next_prime(10^99)
sage: K.<a> = NumberField(x^3 - p*q)
sage: K.maximal_order(prime_range(10000)).basis()
[1, a, a^2]
optimized_representation ( names=None, both_maps=True)
Return a field isomorphic to self with a better defining polynomial if possible, along with field isomorphisms from the new field to self and from self to the new field.
EXAMPLES: We construct a compositum of 3 quadratic fields, then find an optimized representation and
transform elements back and forth.
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sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 +
˓→576
sage: L, from_L, to_L = K.optimized_representation()
sage: L
# your answer may different, since algorithm is random
Number Field in a14 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 + 81
sage: to_L(K.0)
# random
4/189*a14^7 - 1/63*a14^6 + 1/27*a14^5 + 2/9*a14^4 - 5/27*a14^3 + 8/9*a14^2 +
˓→3/7*a14 + 3/7
sage: from_L(L.0)
# random
1/1152*a1^7 + 1/192*a1^6 + 23/576*a1^5 + 17/96*a1^4 + 37/72*a1^3 + 5/6*a1^2 +
˓→55/24*a1 + 3/4
The transformation maps are mutually inverse isomorphisms.
sage: from_L(to_L(K.0))
b
sage: to_L(from_L(L.0))
a14
# random
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage: K.<a> = NumberField(7/9*x^3 + 7/3*x^2 - 56*x + 123)
sage: K.optimized_representation()
(Number Field in a1 with defining polynomial x^3 - 7*x - 7,
Ring morphism:
From: Number Field in a1 with defining polynomial x^3 - 7*x - 7
To:
Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x
˓→+ 123
Defn: a1 |--> 7/225*a^2 - 7/75*a - 42/25,
Ring morphism:
From: Number Field in a with defining polynomial 7/9*x^3 + 7/3*x^2 - 56*x
˓→+ 123
To:
Number Field in a1 with defining polynomial x^3 - 7*x - 7
Defn: a |--> -15/7*a1^2 + 9)
optimized_subfields ( degree=0, name=None, both_maps=True)
Return optimized representations of many (but not necessarily all!) subfields of self of the given degree,
or of all possible degrees if degree is 0.
EXAMPLES:
sage: K = NumberField([x^2 + p for p in [5, 3, 2]],'a').absolute_field('b'); K
Number Field in b with defining polynomial x^8 + 40*x^6 + 352*x^4 + 960*x^2 +
˓→576
sage: L = K.optimized_subfields(name='b')
sage: L[0][0]
Number Field in b0 with defining polynomial x - 1
sage: L[1][0]
Number Field in b1 with defining polynomial x^2 - 3*x + 3
sage: [z[0] for z in L]
# random -- since algorithm is random
[Number Field in b0 with defining polynomial x - 1,
Number Field in b1 with defining polynomial x^2 - x + 1,
Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25,
Number Field in b3 with defining polynomial x^4 - 2*x^2 + 4,
Number Field in b4 with defining polynomial x^8 + 4*x^6 + 7*x^4 + 36*x^2 +
˓→81]
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We examine one of the optimized subfields in more detail:
sage: M, from_M, to_M = L[2]
sage: M
# random
Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
sage: from_M
# may be slightly random
Ring morphism:
From: Number Field in b2 with defining polynomial x^4 - 5*x^2 + 25
To:
Number Field in a1 with defining polynomial x^8 + 40*x^6 + 352*x^4 +
˓→960*x^2 + 576
Defn: b2 |--> -5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/
˓→72*a1^3 + 17/12*a1^2 - 53/24*a1 - 1
The to_M map is None, since there is no map from K to M:
sage: to_M
We apply the from_M map to the generator of M, which gives a rather large element of 𝐾:
sage: from_M(M.0)
# random
-5/1152*a1^7 + 1/96*a1^6 - 97/576*a1^5 + 17/48*a1^4 - 95/72*a1^3 + 17/12*a1^2
˓→- 53/24*a1 - 1
Nevertheless, that large-ish element lies in a degree 4 subfield:
sage: from_M(M.0).minpoly()
x^4 - 5*x^2 + 25
# random
order ( *args, **kwds)
Return the order with given ring generators in the maximal order of this number field.
INPUT:
•gens - list of elements in this number field; if no generators are given, just returns the cardinality of
this number field (∞) for consistency.
•check_is_integral - bool (default: True ), whether to check that each generator is integral.
•check_rank - bool (default: True ), whether to check that the ring generated by gens is of full
rank.
•allow_subfield - bool (default: False ), if True and the generators do not generate an order,
i.e., they generate a subring of smaller rank, instead of raising an error, return an order in a smaller
number field.
EXAMPLES:
sage: k.<i> = NumberField(x^2 + 1)
sage: k.order(2*i)
Order in Number Field in i with defining polynomial x^2 + 1
sage: k.order(10*i)
Order in Number Field in i with defining polynomial x^2 + 1
sage: k.order(3)
Traceback (most recent call last):
...
ValueError: the rank of the span of gens is wrong
sage: k.order(i/2)
Traceback (most recent call last):
...
ValueError: each generator must be integral
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Alternatively, an order can be constructed by adjoining elements to Z:
sage: K.<a> = NumberField(x^3 - 2)
sage: ZZ[a]
Order in Number Field in a0 with defining polynomial x^3 - 2
places ( all_complex=False, prec=None)
Return the collection of all infinite places of self.
By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one
of each pair of complex conjugate homomorphisms into CIF.
On the other hand, if prec is not None, we simply return places into RealField(prec) and ComplexField(prec) (or RDF, CDF if prec=53). One can also use prec=infinity , which returns embeddings
into the field Q of algebraic numbers (or its subfield A of algebraic reals); this permits exact computation,
but can be extremely slow.
There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real
embeddings are returned as embeddings into CIF instead of RIF.
EXAMPLES:
sage: F.<alpha> = NumberField(x^3-100*x+1) ; F.places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:
Real Field with 106 bits of precision
Defn: alpha |--> -10.00499625499181184573367219280,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:
Real Field with 106 bits of precision
Defn: alpha |--> 0.01000001000003000012000055000273,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 - 100*x + 1
To:
Real Field with 106 bits of precision
Defn: alpha |--> 9.994996244991781845613530439509]
sage: F.<alpha> = NumberField(x^3+7) ; F.places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Real Field with 106 bits of precision
Defn: alpha |--> -1.912931182772389101199116839549,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Complex Field with 53 bits of precision
Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
sage: F.<alpha> = NumberField(x^3+7) ; F.places(all_complex=True)
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Complex Field with 53 bits of precision
Defn: alpha |--> -1.91293118277239,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Complex Field with 53 bits of precision
Defn: alpha |--> 0.956465591386195 + 1.65664699997230*I]
sage: F.places(prec=10)
[Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Real Field with 10 bits of precision
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Defn: alpha |--> -1.9,
Ring morphism:
From: Number Field in alpha with defining polynomial x^3 + 7
To:
Complex Field with 10 bits of precision
Defn: alpha |--> 0.96 + 1.7*I]
real_places ( prec=None)
Return all real places of self as homomorphisms into RIF.
EXAMPLES:
sage: F.<alpha> = NumberField(x^4-7) ; F.real_places()
[Ring morphism:
From: Number Field in alpha with defining polynomial x^4 - 7
To:
Real Field with 106 bits of precision
Defn: alpha |--> -1.626576561697785743211232345494,
Ring morphism:
From: Number Field in alpha with defining polynomial x^4 - 7
To:
Real Field with 106 bits of precision
Defn: alpha |--> 1.626576561697785743211232345494]
relative_degree ( )
A synonym for degree.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_degree()
2
relative_different ( )
A synonym for different.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_different()
Fractional ideal (2)
relative_discriminant ( )
A synonym for discriminant.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_discriminant()
-4
relative_polynomial ( )
A synonym for polynomial.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_polynomial()
x^2 + 1
relative_vector_space ( )
A synonym for vector_space.
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EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.relative_vector_space()
(Vector space of dimension 2 over Rational Field,
Isomorphism map:
From: Vector space of dimension 2 over Rational Field
To:
Number Field in i with defining polynomial x^2 + 1,
Isomorphism map:
From: Number Field in i with defining polynomial x^2 + 1
To:
Vector space of dimension 2 over Rational Field)
relativize ( alpha, names, structure=None)
Given an element in self or an embedding of a subfield into self, return a relative number field 𝐾 isomorphic to self that is relative over the absolute field Q(𝛼) or the domain of 𝑎𝑙𝑝ℎ𝑎, along with isomorphisms
from 𝐾 to self and from self to 𝐾.
INPUT:
•alpha - an element of self or an embedding of a subfield into self
•names - 2-tuple of names of generator for output field K and the subfield QQ(alpha) names[0]
generators K and names[1] QQ(alpha).
•structure – an instance of structure.NumberFieldStructure or None (default: None
), if None , then the resulting field’s structure() will return isomorphisms from and to this field.
Otherwise, the field will be equipped with structure .
OUTPUT:
K – relative number field
Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and
to_K is an isomorphism from self to K.
EXAMPLES:
sage: K.<a> = NumberField(x^10 - 2)
sage: L.<c,d> = K.relativize(a^4 + a^2 + 2); L
Number Field in c with defining polynomial x^2 - 1/5*d^4 + 8/5*d^3 - 23/5*d^2
˓→+ 7*d - 18/5 over its base field
sage: c.absolute_minpoly()
x^10 - 2
sage: d.absolute_minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: (a^4 + a^2 + 2).minpoly()
x^5 - 10*x^4 + 40*x^3 - 90*x^2 + 110*x - 58
sage: from_L, to_L = L.structure()
sage: to_L(a)
c
sage: to_L(a^4 + a^2 + 2)
d
sage: from_L(to_L(a^4 + a^2 + 2))
a^4 + a^2 + 2
The following demonstrates distinct embeddings of a subfield into a larger field:
sage: K.<a> = NumberField(x^4 + 2*x^2 + 2)
sage: K0 = K.subfields(2)[0][0]; K0
Number Field in a0 with defining polynomial x^2 - 2*x + 2
sage: rho, tau = K0.embeddings(K)
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sage: L0 = K.relativize(rho(K0.gen()), 'b'); L0
Number Field in b0 with defining polynomial x^2 - b1 + 2 over its base field
sage: L1 = K.relativize(rho, 'b'); L1
Number Field in b with defining polynomial x^2 - a0 + 2 over its base field
sage: L2 = K.relativize(tau, 'b'); L2
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: L0.base_field() is K0
False
sage: L1.base_field() is K0
True
sage: L2.base_field() is K0
True
Here we see that with the different embeddings, the relative norms are different:
sage:
sage:
sage:
sage:
4
sage:
sage:
True
sage:
True
a0 = K0.gen()
L1_into_K, K_into_L1 = L1.structure()
L2_into_K, K_into_L2 = L2.structure()
len(K.factor(41))
w1 = -a^2 + a + 1; P = K.ideal([w1])
Pp = L1.ideal(K_into_L1(w1)).ideal_below(); Pp == K0.ideal([4*a0 + 1])
Pp == w1.norm(rho)
sage: w2 = a^2 + a - 1; Q = K.ideal([w2])
sage: Qq = L2.ideal(K_into_L2(w2)).ideal_below(); Qq == K0.ideal([-4*a0 + 9])
True
sage: Qq == w2.norm(tau)
True
sage: Pp == Qq
False
subfields ( degree=0, name=None)
Return all subfields of self of the given degree, or of all possible degrees if degree is 0. The subfields are
returned as absolute fields together with an embedding into self. For the case of the field itself, the reverse
isomorphism is also provided.
EXAMPLES:
sage: K.<a> = NumberField( [x^3 - 2, x^2 + x + 1] )
sage: K = K.absolute_field('b')
sage: S = K.subfields()
sage: len(S)
6
sage: [k[0].polynomial() for k in S]
[x - 3,
x^2 - 3*x + 9,
x^3 - 3*x^2 + 3*x + 1,
x^3 - 3*x^2 + 3*x + 1,
x^3 - 3*x^2 + 3*x - 17,
x^6 - 3*x^5 + 6*x^4 - 11*x^3 + 12*x^2 + 3*x + 1]
sage: R.<t> = QQ[]
sage: L = NumberField(t^3 - 3*t + 1, 'c')
sage: [k[1] for k in L.subfields()]
[Ring morphism:
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From: Number Field in c0 with defining polynomial t
To:
Number Field in c with defining polynomial t^3 - 3*t + 1
Defn: 0 |--> 0,
Ring morphism:
From: Number Field in c1 with defining polynomial t^3 - 3*t + 1
To:
Number Field in c with defining polynomial t^3 - 3*t + 1
Defn: c1 |--> c]
sage: len(L.subfields(2))
0
sage: len(L.subfields(1))
1
vector_space ( )
Return a vector space V and isomorphisms self –> V and V –> self.
OUTPUT:
•V - a vector space over the rational numbers
•from_V - an isomorphism from V to self
•to_V - an isomorphism from self to V
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 2)
sage: V, from_V, to_V = k.vector_space()
sage: from_V(V([1,2,3]))
3*a^2 + 2*a + 1
sage: to_V(1 + 2*a + 3*a^2)
(1, 2, 3)
sage: V
Vector space of dimension 3 over Rational Field
sage: to_V
Isomorphism map:
From: Number Field in a with defining polynomial x^3 + 2
To:
Vector space of dimension 3 over Rational Field
sage: from_V(to_V(2/3*a - 5/8))
2/3*a - 5/8
sage: to_V(from_V(V([0,-1/7,0])))
(0, -1/7, 0)
sage.rings.number_field.number_field. NumberField_absolute_v1 ( poly,
name,
latex_name,
canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field import NumberField_absolute_v1
sage: R.<x> = QQ[]
sage: NumberField_absolute_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1
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class sage.rings.number_field.number_field. NumberField_cyclotomic ( n,
names,
embedding=None,
assume_disc_small=False,
maximize_at_primes=None)
Bases: sage.rings.number_field.number_field.NumberField_absolute
Create a cyclotomic extension of the rational field.
The command CyclotomicField(n) creates the n-th cyclotomic field, obtained by adjoining an n-th root of unity
to the rational field.
EXAMPLES:
sage: CyclotomicField(3)
Cyclotomic Field of order 3 and degree 2
sage: CyclotomicField(18)
Cyclotomic Field of order 18 and degree 6
sage: z = CyclotomicField(6).gen(); z
zeta6
sage: z^3
-1
sage: (1+z)^3
6*zeta6 - 3
sage: K = CyclotomicField(197)
sage: loads(K.dumps()) == K
True
sage: loads((z^2).dumps()) == z^2
True
sage: cf12 = CyclotomicField( 12 )
sage: z12 = cf12.0
sage: cf6 = CyclotomicField( 6 )
sage: z6 = cf6.0
sage: FF = Frac( cf12['x'] )
sage: x = FF.0
sage: z6*x^3/(z6 + x)
zeta12^2*x^3/(x + zeta12^2)
sage: cf6 = CyclotomicField(6) ; z6 = cf6.gen(0)
sage: cf3 = CyclotomicField(3) ; z3 = cf3.gen(0)
sage: cf3(z6)
zeta3 + 1
sage: cf6(z3)
zeta6 - 1
sage: type(cf6(z3))
<type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_
˓→quadratic'>
sage: cf1 = CyclotomicField(1) ; z1 = cf1.0
sage: cf3(z1)
1
sage: type(cf3(z1))
<type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_
˓→quadratic'>
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complex_embedding ( prec=53)
Return the embedding of this cyclotomic field into the approximate complex field with precision prec
obtained by sending the generator 𝜁 of self to exp(2*pi*i/n), where 𝑛 is the multiplicative order of 𝜁.
EXAMPLES:
sage: C = CyclotomicField(4)
sage: C.complex_embedding()
Ring morphism:
From: Cyclotomic Field of order 4 and degree 2
To:
Complex Field with 53 bits of precision
Defn: zeta4 |--> 6.12323399573677e-17 + 1.00000000000000*I
Note in the example above that the way zeta is computed (using sin and cosine in MPFR) means that only
the prec bits of the number after the decimal point are valid.
sage:
sage:
sage:
-0.50
sage:
1.0
sage:
0.00
sage:
8.0
K = CyclotomicField(3)
phi = K.complex_embedding(10)
phi(K.0)
+ 0.87*I
phi(K.0^3)
phi(K.0^3 - 1)
phi(K.0^3 + 7)
complex_embeddings ( prec=53)
Return all embeddings of this cyclotomic field into the approximate complex field with precision prec.
If you want 53-bit double precision, which is faster but less reliable, then do self.embeddings(CDF)
.
EXAMPLES:
sage: CyclotomicField(5).complex_embeddings()
[
Ring morphism:
From: Cyclotomic Field of order 5 and degree 4
To:
Complex Field with 53 bits of precision
Defn: zeta5 |--> 0.309016994374947 + 0.951056516295154*I,
Ring morphism:
From: Cyclotomic Field of order 5 and degree 4
To:
Complex Field with 53 bits of precision
Defn: zeta5 |--> -0.809016994374947 + 0.587785252292473*I,
Ring morphism:
From: Cyclotomic Field of order 5 and degree 4
To:
Complex Field with 53 bits of precision
Defn: zeta5 |--> -0.809016994374947 - 0.587785252292473*I,
Ring morphism:
From: Cyclotomic Field of order 5 and degree 4
To:
Complex Field with 53 bits of precision
Defn: zeta5 |--> 0.309016994374947 - 0.951056516295154*I
]
construction ( )
Return data defining a functorial construction of self .
EXAMPLES:
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sage: F, R = CyclotomicField(5).construction()
sage: R
Rational Field
sage: F.polys
[x^4 + x^3 + x^2 + x + 1]
sage: F.names
['zeta5']
sage: F.embeddings
[0.309016994374948? + 0.951056516295154?*I]
sage: F.structures
[None]
different ( )
Returns the different ideal of the cyclotomic field self.
EXAMPLES:
sage: C20 = CyclotomicField(20)
sage: C20.different()
Fractional ideal (10, 2*zeta20^6 - 4*zeta20^4 - 4*zeta20^2 + 2)
sage: C18 = CyclotomicField(18)
sage: D = C18.different().norm()
sage: D == C18.discriminant().abs()
True
discriminant ( v=None)
Returns the discriminant of the ring of integers of the cyclotomic field self, or if v is specified, the determinant of the trace pairing on the elements of the list v.
Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the
discriminant of composita.
INPUT:
•v (optional) - list of element of this number field
OUTPUT: Integer if v is omitted, and Rational otherwise.
EXAMPLES:
sage: CyclotomicField(20).discriminant()
4000000
sage: CyclotomicField(18).discriminant()
-19683
is_galois ( )
Return True since all cyclotomic fields are automatically Galois.
EXAMPLES:
sage: CyclotomicField(29).is_galois()
True
is_isomorphic ( other)
Return True if the cyclotomic field self is isomorphic as a number field to other.
EXAMPLES:
sage: CyclotomicField(11).is_isomorphic(CyclotomicField(22))
True
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sage: CyclotomicField(11).is_isomorphic(CyclotomicField(23))
False
sage: CyclotomicField(3).is_isomorphic(NumberField(x^2 + x +1, 'a'))
True
sage: CyclotomicField(18).is_isomorphic(CyclotomicField(9))
True
sage: CyclotomicField(10).is_isomorphic(NumberField(x^4 - x^3 + x^2 - x + 1,
˓→'b'))
True
Check trac ticket #14300:
sage:
sage:
sage:
False
sage:
False
K = CyclotomicField(4)
N = K.extension(x^2-5, 'z')
K.is_isomorphic(N)
K.is_isomorphic(CyclotomicField(8))
next_split_prime ( p=2)
Return the next prime integer 𝑝 that splits completely in this cyclotomic field (and does not ramify).
EXAMPLES:
sage: K.<z> = CyclotomicField(3)
sage: K.next_split_prime(7)
13
number_of_roots_of_unity ( )
Return number of roots of unity in this cyclotomic field.
EXAMPLES:
sage: K.<a> = CyclotomicField(21)
sage: K.number_of_roots_of_unity()
42
real_embeddings ( prec=53)
Return all embeddings of this cyclotomic field into the approximate real field with precision prec.
Mostly, of course, there are no such embeddings.
EXAMPLES:
sage: CyclotomicField(4).real_embeddings()
[]
sage: CyclotomicField(2).real_embeddings()
[
Ring morphism:
From: Cyclotomic Field of order 2 and degree 1
To:
Real Field with 53 bits of precision
Defn: -1 |--> -1.00000000000000
]
roots_of_unity ( )
Return all the roots of unity in this cyclotomic field, primitive or not.
EXAMPLES:
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sage: K.<a> = CyclotomicField(3)
sage: zs = K.roots_of_unity(); zs
[1, a, -a - 1, -1, -a, a + 1]
sage: [ z**K.number_of_roots_of_unity() for z in zs ]
[1, 1, 1, 1, 1, 1]
signature ( )
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of
this cyclotomic field, respectively.
Trivial since, apart from QQ, cyclotomic fields are totally complex.
EXAMPLES:
sage: CyclotomicField(5).signature()
(0, 2)
sage: CyclotomicField(2).signature()
(1, 0)
zeta ( n=None, all=False)
Return an element of multiplicative order 𝑛 in this this cyclotomic field. If there is no such element, raise
a ValueError .
INPUT:
•n – integer (default: None, returns element of maximal order)
•all – bool (default: False) - whether to return a list of all primitive 𝑛-th roots of unity.
OUTPUT: root of unity or list
EXAMPLES:
sage:
sage:
zeta4
sage:
-1
sage:
4
k = CyclotomicField(4)
k.zeta()
k.zeta(2)
k.zeta().multiplicative_order()
sage: k = CyclotomicField(21)
sage: k.zeta().multiplicative_order()
42
sage: k.zeta(21).multiplicative_order()
21
sage: k.zeta(7).multiplicative_order()
7
sage: k.zeta(6).multiplicative_order()
6
sage: k.zeta(84)
Traceback (most recent call last):
..
ValueError: 84 does not divide order of generator (42)
sage: K.<a> = CyclotomicField(7)
sage: K.zeta(all=True)
[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
sage: K.zeta(14, all=True)
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[-a^4, -a^5, a^5 + a^4 + a^3 + a^2 + a + 1, -a, -a^2, -a^3]
sage: K.zeta(2, all=True)
[-1]
sage: K.<a> = CyclotomicField(10)
sage: K.zeta(20, all=True)
Traceback (most recent call last):
...
ValueError: 20 does not divide order of generator (10)
sage: K.<a> = CyclotomicField(5)
sage: K.zeta(4)
Traceback (most recent call last):
...
ValueError: 4 does not divide order of generator (10)
sage: v = K.zeta(5, all=True); v
[a, a^2, a^3, -a^3 - a^2 - a - 1]
sage: [b^5 for b in v]
[1, 1, 1, 1]
zeta_order ( )
Return the order of the maximal root of unity contained in this cyclotomic field.
EXAMPLES:
sage:
2
sage:
4
sage:
10
sage:
5
sage:
778
CyclotomicField(1).zeta_order()
CyclotomicField(4).zeta_order()
CyclotomicField(5).zeta_order()
CyclotomicField(5)._n()
CyclotomicField(389).zeta_order()
sage.rings.number_field.number_field. NumberField_cyclotomic_v1 ( zeta_order,
name, canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field import NumberField_cyclotomic_v1
sage: NumberField_cyclotomic_v1(5,'a')
Cyclotomic Field of order 5 and degree 4
sage: NumberField_cyclotomic_v1(5,'a').variable_name()
'a'
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class sage.rings.number_field.number_field. NumberField_generic ( polynomial,
name,
latex_name,
check=True,
embedding=None, category=None, assume_disc_small=False,
maximize_at_primes=None,
structure=None)
Bases: sage.misc.fast_methods.WithEqualityById , sage.rings.number_field.number_field_base.
Generic class for number fields defined by an irreducible polynomial over Q.
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2); K
Number Field in a with defining polynomial x^3 - 2
sage: TestSuite(K).run()
S_class_group ( S, proof=None, names=’c’)
Returns the S-class group of this number field over its base field.
INPUT:
•S - a set of primes of the base field
•proof - if False, assume the GRH in computing the class group.
number_field_proof to change this default globally.
Default is True.
Call
•names - names of the generators of this class group.
OUTPUT:
The S-class group of this number field.
EXAMPLES:
A well known example:
sage: K.<a> = QuadraticField(-5)
sage: K.S_class_group([])
S-class group of order 2 with structure C2 of Number Field in a with defining
˓→polynomial x^2 + 5
When we include the prime (2, 𝑎 + 1), the S-class group becomes trivial:
sage: K.S_class_group([K.ideal(2,a+1)])
S-class group of order 1 of Number Field in a with defining polynomial x^2 + 5
S_unit_group ( proof=None, S=None)
Return the S-unit group (including torsion) of this number field.
ALGORITHM: Uses PARI’s pari:bnfsunit command.
INPUT:
•proof (bool, default True) flag passed to pari .
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•S - list or tuple of prime ideals, or an ideal, or a single ideal or element from which an ideal can
be constructed, in which case the support is used. If None, the global unit group is constructed;
otherwise, the S-unit group is constructed.
Note: The group is cached.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3)
sage: U = K.S_unit_group(S=a); U
S-unit group with structure C10 x Z x Z x Z of Number Field in a with
˓→defining polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375 with S =
˓→(Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5), Fractional ideal
˓→(11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
sage: U.gens()
(u0, u1, u2, u3)
sage: U.gens_values() # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3, 1/
˓→275*a^3 + 4/55*a^2 - 5/11*a + 5, -14/275*a^3 + 21/55*a^2 - 29/11*a + 6]
sage: U.invariants()
(10, 0, 0, 0)
sage: [u.multiplicative_order() for u in U.gens()]
[10, +Infinity, +Infinity, +Infinity]
sage: U.primes()
(Fractional ideal (5, 1/275*a^3 + 4/55*a^2 - 5/11*a + 5), Fractional ideal
˓→(11, 1/275*a^3 + 4/55*a^2 - 5/11*a + 9))
With the default value of 𝑆, the S-unit group is the same as the global unit group:
sage:
sage:
sage:
sage:
True
x = polygen(QQ)
K.<a> = NumberField(x^3 + 3)
U = K.unit_group(proof=False)
U == K.S_unit_group(proof=False)
The value of 𝑆 may be specified as a list of prime ideals, or an ideal, or an element of the field:
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=K.ideal(6).prime_factors()); U
S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with
˓→defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1),
˓→Fractional ideal (a + 1), Fractional ideal (a))
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=K.ideal(6)); U
S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with
˓→defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1),
˓→Fractional ideal (a + 1), Fractional ideal (a))
sage: K.<a> = NumberField(x^3 + 3)
sage: U = K.S_unit_group(proof=False, S=6); U
S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with
˓→defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1),
˓→Fractional ideal (a + 1), Fractional ideal (a))
sage: U
S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with
˓→defining polynomial x^3 + 3 with S = (Fractional ideal (-a^2 + a - 1),
˓→Fractional ideal (a + 1), Fractional ideal (a))
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sage: U.primes()
(Fractional ideal (-a^2 + a - 1),
Fractional ideal (a + 1),
Fractional ideal (a))
sage: U.gens()
(u0, u1, u2, u3, u4)
sage: U.gens_values()
[-1, a^2 - 2, -a^2 + a - 1, a + 1, a]
The exp and log methods can be used to create 𝑆-units from sequences of exponents, and recover the
exponents:
sage: U.gens_orders()
(2, 0, 0, 0, 0)
sage: u = U.exp((3,1,4,1,5)); u
-6*a^2 + 18*a - 54
sage: u.norm().factor()
-1 * 2^9 * 3^5
sage: U.log(u)
(1, 1, 4, 1, 5)
S_units ( S, proof=True)
Returns a list of generators of the S-units.
INPUT:
•S – a set of primes of the base field
•proof – if False , assume the GRH in computing the class group
OUTPUT:
A list of generators of the unit group.
Note:
For more functionality see the S_unit_group() function.
EXAMPLES:
sage: K.<a> = QuadraticField(-3)
sage: K.unit_group()
Unit group with structure C6 of Number Field in a with defining polynomial x^
˓→2 + 3
sage: K.S_units([]) # random
[1/2*a + 1/2]
sage: K.S_units([])[0].multiplicative_order()
6
An example in a relative extension (see trac ticket #8722):
sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: W = L.S_units([p]); [x.norm() for x in W]
[9, 1, 1]
Our generators should have the correct parent (trac ticket #9367):
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sage: _.<x> = QQ[]
sage: L.<alpha> = NumberField(x^3 + x + 1)
sage: p = L.S_units([ L.ideal(7) ])
sage: p[0].parent()
Number Field in alpha with defining polynomial x^3 + x + 1
absolute_degree ( )
Return the degree of self over Q.
EXAMPLES:
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').absolute_degree()
3
sage: NumberField(x + 1, 'a').absolute_degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).absolute_degree()
997
absolute_field ( names)
Returns self as an absolute extension over QQ.
OUTPUT:
•K - this number field (since it is already absolute)
Also, K.structure() returns from_K and to_K, where from_K is an isomorphism from K to self and
to_K is an isomorphism from self to K.
EXAMPLES:
sage: K = CyclotomicField(5)
sage: K.absolute_field('a')
Number Field in a with defining polynomial x^4 + x^3 + x^2 + x + 1
absolute_polynomial_ntl ( )
Alias for polynomial_ntl() . Mostly for internal use.
EXAMPLES:
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()
([-27 34 51], 51)
algebraic_closure ( )
Return the algebraic closure of self (which is QQbar).
EXAMPLES:
sage: K.<i> = QuadraticField(-1)
sage: K.algebraic_closure()
Algebraic Field
sage: K.<a> = NumberField(x^3-2)
sage: K.algebraic_closure()
Algebraic Field
sage: K = CyclotomicField(23)
sage: K.algebraic_closure()
Algebraic Field
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change_generator ( alpha, name=None, names=None)
Given the number field self, construct another isomorphic number field 𝐾 generated by the element alpha
of self, along with isomorphisms from 𝐾 to self and from self to 𝐾.
EXAMPLES:
sage: L.<i> = NumberField(x^2 + 1); L
Number Field in i with defining polynomial x^2 + 1
sage: K, from_K, to_K = L.change_generator(i/2 + 3)
sage: K
Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
sage: from_K
Ring morphism:
From: Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
To:
Number Field in i with defining polynomial x^2 + 1
Defn: i0 |--> 1/2*i + 3
sage: to_K
Ring morphism:
From: Number Field in i with defining polynomial x^2 + 1
To:
Number Field in i0 with defining polynomial x^2 - 6*x + 37/4
Defn: i |--> 2*i0 - 6
We can also do
sage: K.<c>, from_K, to_K = L.change_generator(i/2 + 3); K
Number Field in c with defining polynomial x^2 - 6*x + 37/4
We compute the image of the generator
√
−1 of 𝐿.
sage: to_K(i)
2*c - 6
Note that the image is indeed a square root of -1.
sage: to_K(i)^2
-1
sage: from_K(to_K(i))
i
sage: to_K(from_K(c))
c
characteristic ( )
Return the characteristic of this number field, which is of course 0.
EXAMPLES:
sage: k.<a> = NumberField(x^99 + 2); k
Number Field in a with defining polynomial x^99 + 2
sage: k.characteristic()
0
class_group ( proof=None, names=’c’)
Return the class group of the ring of integers of this number field.
INPUT:
•proof - if True then compute the class group provably correctly. Default is True. Call number_field_proof to change this default globally.
•names - names of the generators of this class group.
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OUTPUT: The class group of this number field.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining
˓→polynomial x^2 + 23
sage: G.0
Fractional ideal class (2, 1/2*a - 1/2)
sage: G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)
sage: G.number_field()
Number Field in a with defining polynomial x^2 + 23
sage: G is K.class_group()
True
sage: G is K.class_group(proof=False)
False
sage: G.gens()
(Fractional ideal class (2, 1/2*a - 1/2),)
There can be multiple generators:
sage: k.<a> = NumberField(x^2 + 20072)
sage: G = k.class_group(); G
Class group of order 76 with structure C38 x C2 of Number Field in a with
˓→defining polynomial x^2 + 20072
sage: G.0 # random
Fractional ideal class (41, a + 10)
sage: G.0^38
Trivial principal fractional ideal class
sage: G.1 # random
Fractional ideal class (2, -1/2*a)
sage: G.1^2
Trivial principal fractional ideal class
Class groups of Hecke polynomials tend to be very small:
sage: f = ModularForms(97, 2).T(2).charpoly()
sage: f.factor()
(x - 3) * (x^3 + 4*x^2 + 3*x - 1) * (x^4 - 3*x^3 - x^2 + 6*x - 1)
sage: [NumberField(g,'a').class_group().order() for g,_ in f.factor()]
[1, 1, 1]
class_number ( proof=None)
Return the class number of this number field, as an integer.
INPUT:
•proof - bool (default: True unless you called number_field_proof)
EXAMPLES:
sage: NumberField(x^2 + 23, 'a').class_number()
3
sage: NumberField(x^2 + 163, 'a').class_number()
1
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').class_number(proof=False)
1539
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completion ( p, prec, extras={})
Returns the completion of self at 𝑝 to the specified precision. Only implemented at archimedean places,
and then only if an embedding has been fixed.
EXAMPLES:
sage: K.<a> = QuadraticField(2)
sage: K.completion(infinity, 100)
Real Field with 100 bits of precision
sage: K.<zeta> = CyclotomicField(12)
sage: K.completion(infinity, 53, extras={'type': 'RDF'})
Complex Double Field
sage: zeta + 1.5
# implicit test
2.36602540378444 + 0.500000000000000*I
complex_conjugation ( )
Return the complex conjugation of self.
This is only well-defined for fields contained in CM fields (i.e. for totally real fields and CM fields).
Recall that a CM field is a totally imaginary quadratic extension of a totally real field. For other fields, a
ValueError is raised.
EXAMPLES:
sage: QuadraticField(-1, 'I').complex_conjugation()
Ring endomorphism of Number Field in I with defining polynomial x^2 + 1
Defn: I |--> -I
sage: CyclotomicField(8).complex_conjugation()
Ring endomorphism of Cyclotomic Field of order 8 and degree 4
Defn: zeta8 |--> -zeta8^3
sage: QuadraticField(5, 'a').complex_conjugation()
Identity endomorphism of Number Field in a with defining polynomial x^2 - 5
sage: F = NumberField(x^4 + x^3 - 3*x^2 - x + 1, 'a')
sage: F.is_totally_real()
True
sage: F.complex_conjugation()
Identity endomorphism of Number Field in a with defining polynomial x^4 + x^3
˓→- 3*x^2 - x + 1
sage: F.<b> = NumberField(x^2 - 2)
sage: F.extension(x^2 + 1, 'a').complex_conjugation()
Relative number field endomorphism of Number Field in a with defining
˓→polynomial x^2 + 1 over its base field
Defn: a |--> -a
b |--> b
sage: F2.<b> = NumberField(x^2 + 2)
sage: K2.<a> = F2.extension(x^2 + 1)
sage: cc = K2.complex_conjugation()
sage: cc(a)
-a
sage: cc(b)
-b
complex_embeddings ( prec=53)
Return all homomorphisms of this number field into the approximate complex field with precision prec.
This always embeds into an MPFR based complex field. If you want embeddings into the 53-bit double
precision, which is faster, use self.embeddings(CDF) .
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EXAMPLES:
sage: k.<a> = NumberField(x^5 + x + 17)
sage: v = k.complex_embeddings()
sage: ls = [phi(k.0^2) for phi in v] ; ls # random order
[2.97572074038...,
-2.40889943716 + 1.90254105304*I,
-2.40889943716 - 1.90254105304*I,
0.921039066973 + 3.07553311885*I,
0.921039066973 - 3.07553311885*I]
sage: K.<a> = NumberField(x^3 + 2)
sage: ls = K.complex_embeddings() ; ls # random order
[
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Complex Double Field
Defn: a |--> -1.25992104989...,
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Complex Double Field
Defn: a |--> 0.629960524947 - 1.09112363597*I,
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Complex Double Field
Defn: a |--> 0.629960524947 + 1.09112363597*I
]
composite_fields ( other, names=None, both_maps=False, preserve_embedding=True)
Return the possible composite number fields formed from self and other .
INPUT:
•other – number field
•names – generator name for composite fields
•both_maps – boolean (default: False )
•preserve_embedding – boolean (default: True)
OUTPUT:
A list of the composite fields, possibly with maps.
If both_maps is True , the list consists of quadruples (F,self_into_F,other_into_F,k)
such that self_into_F is an embedding of self in F , other_into_F is an embedding of in F ,
and k is one of the following:
•an integer such that F.gen()
k*self_into_F(self.gen()) ;
equals
other_into_F(other.gen()) +
•Infinity , in which case F.gen() equals self_into_F(self.gen()) ;
•None (when other is a relative number field).
If both self and other have embeddings into an ambient field, then each F will have an embedding
with respect to which both self_into_F and other_into_F will be compatible with the ambient
embeddings.
If preserve_embedding is True and if self and other both have embeddings into the same
ambient field, or into fields which are contained in a common field, only the compositum respecting both
embeddings is returned. In all other cases, all possible composite number fields are returned.
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EXAMPLES:
sage: K.<a> = NumberField(x^4 - 2)
sage: K.composite_fields(K)
[Number Field in a with defining polynomial x^4 - 2,
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]
A particular compositum is selected, together with compatible maps into the compositum, if the fields are
endowed with a real or complex embedding:
sage: K1 = NumberField(x^4 - 2, 'a', embedding=RR(2^(1/4)))
sage: K2 = NumberField(x^4 - 2, 'a', embedding=RR(-2^(1/4)))
sage: K1.composite_fields(K2)
[Number Field in a with defining polynomial x^4 - 2]
sage: [F, f, g, k], = K1.composite_fields(K2, both_maps=True); F
Number Field in a with defining polynomial x^4 - 2
sage: f(K1.0), g(K2.0)
(a, -a)
With preserve_embedding set to False , the embeddings are ignored:
sage: K1.composite_fields(K2, preserve_embedding=False)
[Number Field in a with defining polynomial x^4 - 2,
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500]
Changing the embedding selects a different compositum:
sage: K3 = NumberField(x^4 - 2, 'a', embedding=CC(2^(1/4)*I))
sage: [F, f, g, k], = K1.composite_fields(K3, both_maps=True); F
Number Field in a0 with defining polynomial x^8 + 28*x^4 + 2500
sage: f(K1.0), g(K3.0)
(1/240*a0^5 - 41/120*a0, 1/120*a0^5 + 19/60*a0)
If no embeddings are specified, the maps into the compositum are chosen arbitrarily:
sage: Q1.<a> = NumberField(x^4 + 10*x^2 + 1)
sage: Q2.<b> = NumberField(x^4 + 16*x^2 + 4)
sage: Q1.composite_fields(Q2, 'c')
[Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 + 3840*x^2
˓→+ 3600]
sage: F, Q1_into_F, Q2_into_F, k = Q1.composite_fields(Q2, 'c', both_
˓→maps=True)[0]
sage: Q1_into_F
Ring morphism:
From: Number Field in a with defining polynomial x^4 + 10*x^2 + 1
To:
Number Field in c with defining polynomial x^8 + 64*x^6 + 904*x^4 +
˓→3840*x^2 + 3600
Defn: a |--> 19/14400*c^7 + 137/1800*c^5 + 2599/3600*c^3 + 8/15*c
This is just one of four embeddings of Q1 into F :
sage: Hom(Q1, F).order()
4
construction ( )
Construction of self
EXAMPLES:
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sage: K.<a>=NumberField(x^3+x^2+1,embedding=CC.gen())
sage: F,R = K.construction()
sage: F
AlgebraicExtensionFunctor
sage: R
Rational Field
The construction functor respects distinguished embeddings:
sage: F(R) is K
True
sage: F.embeddings
[0.2327856159383841? + 0.7925519925154479?*I]
sage:
sage:
sage:
sage:
a*b
P.<x> = QQ[]
K.<a> = NumberField(x^3-5,embedding=0)
L.<b> = K.extension(x^2+a)
a*b
defining_polynomial ( )
Return the defining polynomial of this number field.
This is exactly the same as self.polynomial() .
EXAMPLES:
sage: k5.<z> = CyclotomicField(5)
sage: k5.defining_polynomial()
x^4 + x^3 + x^2 + x + 1
sage: y = polygen(QQ,'y')
sage: k.<a> = NumberField(y^9 - 3*y + 5); k
Number Field in a with defining polynomial y^9 - 3*y + 5
sage: k.defining_polynomial()
y^9 - 3*y + 5
degree ( )
Return the degree of this number field.
EXAMPLES:
sage: NumberField(x^3 + x^2 + 997*x + 1, 'a').degree()
3
sage: NumberField(x + 1, 'a').degree()
1
sage: NumberField(x^997 + 17*x + 3, 'a', check=False).degree()
997
different ( )
Compute the different fractional ideal of this number field.
The codifferent is the fractional ideal of all 𝑥 in 𝐾 such that the trace of 𝑥𝑦 is an integer for all 𝑦 ∈ 𝑂𝐾 .
The different is the integral ideal which is the inverse of the codifferent.
See Wikipedia article Different_ideal
EXAMPLES:
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sage: k.<a> = NumberField(x^2 + 23)
sage: d = k.different()
sage: d
Fractional ideal (-a)
sage: d.norm()
23
sage: k.disc()
-23
The different is cached:
sage: d is k.different()
True
Another example:
sage: k.<b> = NumberField(x^2 - 123)
sage: d = k.different(); d
Fractional ideal (2*b)
sage: d.norm()
492
sage: k.disc()
492
disc ( v=None)
Shortcut for self.discriminant.
EXAMPLES:
sage: k.<b> = NumberField(x^2 - 123)
sage: k.disc()
492
discriminant ( v=None)
Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of
the trace pairing on the elements of the list v.
INPUT:
•v – (optional) list of elements of this number field
OUTPUT:
Integer if 𝑣 is omitted, and Rational otherwise.
EXAMPLES:
sage: K.<t> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.disc()
-503
sage: K.disc([1, t, t^2])
-2012
sage: K.disc([1/7, (1/5)*t, (1/3)*t^2])
-2012/11025
sage: (5*7*3)^2
11025
sage: NumberField(x^2 - 1/2, 'a').discriminant()
8
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elements_of_norm ( n, proof=None)
Return a list of elements of norm n .
INPUT:
•n – integer in this number field
•proof – boolean (default: True , unless you called number_field_proof and set it otherwise)
OUTPUT:
A complete system of integral elements of norm 𝑛, modulo units of positive norm.
EXAMPLES:
sage:
sage:
[]
sage:
[-7*a
K.<a> = NumberField(x^2+1)
K.elements_of_norm(3)
K.elements_of_norm(50)
+ 1, 5*a - 5, 7*a + 1]
extension ( poly, name=None, names=None, *args, **kwds)
Return the relative extension of this field by a given polynomial.
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2)
sage: R.<t> = K[]
sage: L.<b> = K.extension(t^2 + a); L
Number Field in b with defining polynomial t^2 + a over its base field
We create another extension:
sage: k.<a> = NumberField(x^2 +
Number Field in a with defining
sage: y = var('y')
sage: m.<b> = k.extension(y^2 +
Number Field in b with defining
1); k
polynomial x^2 + 1
2); m
polynomial y^2 + 2 over its base field
Note that b is a root of 𝑦 2 + 2:
sage:
x^2 +
sage:
z^2 +
b.minpoly()
2
b.minpoly('z')
2
A relative extension of a relative extension:
sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
sage: R.<z> = k[]
sage: L.<b> = NumberField(z^3 + 3 + a); L
Number Field in b with defining polynomial z^3 + a0 + 3 over its base field
Extension fields with given defining data are unique (trac ticket #20791):
sage: K.<a> = NumberField(x^2 + 1)
sage: K.extension(x^2 - 2, 'b') is K.extension(x^2 - 2, 'b')
True
factor ( n)
Ideal factorization of the principal ideal generated by 𝑛.
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EXAMPLES:
Here we show how to factor Gaussian integers (up to units). First we form a number field defined by
𝑥2 + 1:
sage: K.<I> = NumberField(x^2 + 1); K
Number Field in I with defining polynomial x^2 + 1
Here are the factors:
sage: fi, fj = K.factor(17); fi,fj
((Fractional ideal (I + 4), 1), (Fractional ideal (I - 4), 1))
Now we extract the reduced form of the generators:
sage: zi = fi[0].gens_reduced()[0]; zi
I + 4
sage: zj = fj[0].gens_reduced()[0]; zj
I - 4
We recover the integer that was factored in Z[𝑖] (up to a unit):
sage: zi*zj
-17
One can also factor elements or ideals of the number field:
sage: K.<a> = NumberField(x^2 + 1)
sage: K.factor(1/3)
(Fractional ideal (3))^-1
sage: K.factor(1+a)
Fractional ideal (a + 1)
sage: K.factor(1+a/5)
(Fractional ideal (a + 1)) * (Fractional ideal (-a - 2))^-1 * (Fractional
˓→ideal (2*a + 1))^-1 * (Fractional ideal (-3*a - 2))
An example over a relative number field:
sage: pari('setrand(2)')
sage: L.<b> = K.extension(x^2 - 7)
sage: f = L.factor(a + 1); f
(Fractional ideal (1/2*a*b - a + 1/2)) * (Fractional ideal (-1/2*a*b - a + 1/
˓→2))
sage: f.value() == a+1
True
It doesn’t make sense to factor the ideal (0), so this raises an error:
sage: L.factor(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'factor'
AUTHORS:
•Alex Clemesha (2006-05-20), Francis Clarke (2009-04-21): examples
fractional_ideal ( *gens, **kwds)
Return the ideal in 𝒪𝐾 generated by gens. This overrides the sage.rings.ring.Field method to
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use the sage.rings.ring.Ring one instead, since we’re not really concerned with ideals in a field
but in its ring of integers.
INPUT:
•gens - a list of generators, or a number field ideal.
EXAMPLES:
sage: K.<a> = NumberField(x^3-2)
sage: K.fractional_ideal([1/a])
Fractional ideal (1/2*a^2)
One can also input a number field ideal itself, or, more usefully, for a tower of number fields an ideal in
one of the fields lower down the tower.
sage: K.fractional_ideal(K.ideal(a))
Fractional ideal (a)
sage: L.<b> = K.extension(x^2 - 3, x^2 + 1)
sage: M.<c> = L.extension(x^2 + 1)
sage: L.ideal(K.ideal(2, a))
Fractional ideal (a)
sage: M.ideal(K.ideal(2, a)) == M.ideal(a*(b - c)/2)
True
The zero ideal is not a fractional ideal!
sage: K.fractional_ideal(0)
Traceback (most recent call last):
...
ValueError: gens must have a nonzero element (zero ideal is not a fractional
˓→ideal)
galois_group ( type=None, algorithm=’pari’, names=None)
Return the Galois group of the Galois closure of this number field.
INPUT:
•type - none , gap , or pari . If None (the default), return an explicit group of automorphisms
of self as a GaloisGroup_v2 object. Otherwise, return a GaloisGroup_v1 wrapper object
based on a PARI or Gap transitive group object, which is quicker to compute, but rather less useful
(in particular, it can’t be made to act on self). If type = ‘gap’, the database_gap package should be
installed.
•algorithm - ‘pari’, ‘kash’, ‘magma’. (default: ‘pari’, except when the degree is >= 12 when ‘kash’
is tried.)
•name - a string giving a name for the generator of the Galois closure of self, when self is not Galois.
This is ignored if type is not None.
Note that computing Galois groups as abstract groups is often much faster than computing them
as explicit automorphism groups (but of course you get less information out!) For more (important!) documentation, so the documentation for Galois groups of polynomials over Q, e.g., by typing
K.polynomial().galois_group? , where 𝐾 is a number field.
To obtain actual field homomorphisms from the number field to its splitting field, use type=None.
EXAMPLES:
With type None :
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sage: k.<b> = NumberField(x^2 - 14) # a Galois extension
sage: G = k.galois_group(); G
Galois group of Number Field in b with defining polynomial x^2 - 14
sage: G.gen(0)
(1,2)
sage: G.gen(0)(b)
-b
sage: G.artin_symbol(k.primes_above(3)[0])
(1,2)
sage: k.<b> = NumberField(x^3 - x + 1) # not Galois
sage: G = k.galois_group(names='c'); G
Galois group of Galois closure in c of Number Field in b with defining
˓→polynomial x^3 - x + 1
sage: G.gen(0)
(1,2,3)(4,5,6)
With type 'pari' :
sage: NumberField(x^3-2, 'a').galois_group(type="pari")
Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a
˓→with defining polynomial x^3 - 2
sage: NumberField(x-1, 'a').galois_group(type="gap")
#
˓→gap
Galois group Transitive group number 1 of degree 1 of the
˓→with defining polynomial x - 1
sage: NumberField(x^2+2, 'a').galois_group(type="gap") #
˓→gap
Galois group Transitive group number 1 of degree 2 of the
˓→with defining polynomial x^2 + 2
sage: NumberField(x^3-2, 'a').galois_group(type="gap") #
˓→gap
Galois group Transitive group number 2 of degree 3 of the
˓→with defining polynomial x^3 - 2
optional - database_
Number Field in a
optional - database_
Number Field in a
optional - database_
Number Field in a
sage: x = polygen(QQ)
sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(type='gap')
# optional ˓→ database_gap
Galois group Transitive group number 2 of degree 3 of the Number Field in a
˓→with defining polynomial x^3 + 2*x + 1
sage: NumberField(x^3 + 2*x + 1, 'a').galois_group(algorithm='magma')
#
˓→optional - magma database_gap
Galois group Transitive group number 2 of degree 3 of the Number Field in a
˓→with defining polynomial x^3 + 2*x + 1
EXPLICIT GALOIS GROUP: We compute the Galois group as an explicit group of automorphisms of the
Galois closure of a field.
sage: K.<a> = NumberField(x^3 - 2)
sage: L.<b1> = K.galois_closure(); L
Number Field in b1 with defining polynomial x^6 + 108
sage: G = End(L); G
Automorphism group of Number Field in b1 with defining polynomial x^6 + 108
sage: G.list()
[
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
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Defn: b1 |--> b1,
...
Ring endomorphism of Number Field in b1 with defining polynomial x^6 + 108
Defn: b1 |--> -1/12*b1^4 - 1/2*b1
]
sage: G[2](b1)
1/12*b1^4 + 1/2*b1
gen ( n=0)
Return the generator for this number field.
INPUT:
•n - must be 0 (the default), or an exception is raised.
EXAMPLES:
sage: k.<theta> = NumberField(x^14 + 2); k
Number Field in theta with defining polynomial x^14 + 2
sage: k.gen()
theta
sage: k.gen(1)
Traceback (most recent call last):
...
IndexError: Only one generator.
gen_embedding ( )
If an embedding has been specified, return the image of the generator under that embedding. Otherwise
return None.
EXAMPLES:
sage: QuadraticField(-7, 'a').gen_embedding()
2.645751311064591?*I
sage: NumberField(x^2+7, 'a').gen_embedding() # None
ideal ( *gens, **kwds)
K.ideal() returns a fractional ideal of the field, except for the zero ideal which is not a fractional ideal.
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: K.ideal(2)
Fractional ideal (2)
sage: K.ideal(2+i)
Fractional ideal (i + 2)
sage: K.ideal(0)
Ideal (0) of Number Field in i with defining polynomial x^2 + 1
ideals_of_bdd_norm ( bound)
All integral ideals of bounded norm.
INPUT:
•bound - a positive integer
OUTPUT: A dict of all integral ideals I such that Norm(I) <= bound, keyed by norm.
EXAMPLES:
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sage: K.<a> = NumberField(x^2 + 23)
sage: d = K.ideals_of_bdd_norm(10)
sage: for n in d:
....:
print(n)
....:
for I in d[n]:
....:
print(I)
1
Fractional ideal (1)
2
Fractional ideal (2, 1/2*a - 1/2)
Fractional ideal (2, 1/2*a + 1/2)
3
Fractional ideal (3, 1/2*a - 1/2)
Fractional ideal (3, 1/2*a + 1/2)
4
Fractional ideal (4, 1/2*a + 3/2)
Fractional ideal (2)
Fractional ideal (4, 1/2*a + 5/2)
5
6
Fractional ideal (1/2*a - 1/2)
Fractional ideal (6, 1/2*a + 5/2)
Fractional ideal (6, 1/2*a + 7/2)
Fractional ideal (1/2*a + 1/2)
7
8
Fractional ideal (1/2*a + 3/2)
Fractional ideal (4, a - 1)
Fractional ideal (4, a + 1)
Fractional ideal (1/2*a - 3/2)
9
Fractional ideal (9, 1/2*a + 11/2)
Fractional ideal (3)
Fractional ideal (9, 1/2*a + 7/2)
10
integral_basis ( v=None)
Returns a list containing a ZZ-basis for the full ring of integers of this number field.
INPUT:
•v - None, a prime, or a list of primes. See the documentation for self.maximal_order.
EXAMPLES:
sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.integral_basis()
[1, a, a^2, a^3, a^4]
Next we compute the ring of integers of a cubic field in which 2 is an “essential discriminant divisor”, so
the ring of integers is not generated by a single element.
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: K.integral_basis()
[1, 1/2*a^2 + 1/2*a, a^2]
ALGORITHM: Uses the pari library (via _pari_integral_basis).
is_CM ( )
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Return True if self is a CM field (i.e. a totally imaginary quadratic extension of a totally real field).
EXAMPLES:
sage:
sage:
False
sage:
sage:
True
sage:
sage:
True
sage:
sage:
True
sage:
sage:
False
sage:
sage:
False
sage:
sage:
False
Q.<a> = NumberField(x - 1)
Q.is_CM()
K.<i> = NumberField(x^2 + 1)
K.is_CM()
L.<zeta20> = CyclotomicField(20)
L.is_CM()
K.<omega> = QuadraticField(-3)
K.is_CM()
L.<sqrt5> = QuadraticField(5)
L.is_CM()
F.<a> = NumberField(x^3 - 2)
F.is_CM()
F.<a> = NumberField(x^4-x^3-3*x^2+x+1)
F.is_CM()
The following are non-CM totally imaginary fields.
sage:
sage:
True
sage:
False
sage:
F.<a> = NumberField(x^4 + x^3 - x^2 - x + 1)
F.is_totally_imaginary()
F.is_CM()
F2.<a> = NumberField(x^12 - 5*x^11 + 8*x^10 - 5*x^9 - \
x^8 + 9*x^7 + 7*x^6 - 3*x^5 + 5*x^4 + \
7*x^3 - 4*x^2 - 7*x + 7)
sage: F2.is_totally_imaginary()
True
sage: F2.is_CM()
False
The following is a non-cyclotomic CM field.
sage: M.<a> = NumberField(x^4 - x^3 - x^2 - 2*x + 4)
sage: M.is_CM()
True
Now, we construct a totally imaginary quadratic extension of a totally real field (which is not cyclotomic).
sage: E_0.<a> = NumberField(x^7 - 4*x^6 - 4*x^5 + 10*x^4 + 4*x^3 - \
6*x^2 - x + 1)
sage: E_0.is_totally_real()
True
sage: E.<b> = E_0.extension(x^2 + 1)
sage: E.is_CM()
True
Finally, a CM field that is given as an extension that is not CM.
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sage:
sage:
sage:
sage:
True
sage:
False
E_0.<a> = NumberField(x^2 - 4*x + 16)
y = polygen(E_0)
E.<z> = E_0.extension(y^2 - E_0.gen() / 2)
E.is_CM()
E.is_CM_extension()
is_absolute ( )
Returns True if self is an absolute field.
This function will be implemented in the derived classes.
EXAMPLES:
sage: K = CyclotomicField(5)
sage: K.is_absolute()
True
is_field ( proof=True)
Return True since a number field is a field.
EXAMPLES:
sage: NumberField(x^5 + x + 3, 'c').is_field()
True
is_galois ( )
Return True if this number field is a Galois extension of Q.
EXAMPLES:
sage: NumberField(x^2 + 1, 'i').is_galois()
True
sage: NumberField(x^3 + 2, 'a').is_galois()
False
is_isomorphic ( other)
Return True if self is isomorphic as a number field to other.
EXAMPLES:
54
sage:
sage:
sage:
True
sage:
sage:
False
k.<a> = NumberField(x^2 + 1)
m.<b> = NumberField(x^2 + 4)
k.is_isomorphic(m)
sage:
sage:
True
sage:
True
sage:
False
k = NumberField(x^3 + 2, 'a')
k.is_isomorphic(NumberField((x+1/3)^3 + 2, 'b'))
m.<b> = NumberField(x^2 + 5)
k.is_isomorphic (m)
k.is_isomorphic(NumberField(x^3 + 4, 'b'))
k.is_isomorphic(NumberField(x^3 + 5, 'b'))
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is_relative ( )
EXAMPLES:
sage: K.<a> = NumberField(x^10 - 2)
sage: K.is_absolute()
True
sage: K.is_relative()
False
is_totally_imaginary ( )
Return True if self is totally imaginary, and False otherwise.
Totally imaginary means that no isomorphic embedding of self into the complex numbers has image contained in the real numbers.
EXAMPLES:
sage: NumberField(x^2+2, 'alpha').is_totally_imaginary()
True
sage: NumberField(x^2-2, 'alpha').is_totally_imaginary()
False
sage: NumberField(x^4-2, 'alpha').is_totally_imaginary()
False
is_totally_real ( )
Return True if self is totally real, and False otherwise.
Totally real means that every isomorphic embedding of self into the complex numbers has image contained
in the real numbers.
EXAMPLES:
sage: NumberField(x^2+2, 'alpha').is_totally_real()
False
sage: NumberField(x^2-2, 'alpha').is_totally_real()
True
sage: NumberField(x^4-2, 'alpha').is_totally_real()
False
latex_variable_name ( name=None)
Return the latex representation of the variable name for this number field.
EXAMPLES:
sage: NumberField(x^2 + 3, 'a').latex_variable_name()
'a'
sage: NumberField(x^3 + 3, 'theta3').latex_variable_name()
'\theta_{3}'
sage: CyclotomicField(5).latex_variable_name()
'\zeta_{5}'
maximal_totally_real_subfield ( )
Return the maximal totally real subfield of self together with an embedding of it into self.
EXAMPLES:
sage: F.<a> = QuadraticField(11)
sage: F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^2 - 11, Identity endomorphism
˓→of Number Field in a with defining polynomial x^2 - 11]
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sage: F.<a> = QuadraticField(-15)
sage: F.maximal_totally_real_subfield()
[Rational Field, Natural morphism:
From: Rational Field
To:
Number Field in a with defining polynomial x^2 + 15]
sage: F.<a> = CyclotomicField(29)
sage: F.maximal_totally_real_subfield()
(Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 - 12*x^11
˓→+ 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^4 - 56*x^
˓→3 + 28*x^2 + 7*x - 1, Ring morphism:
From: Number Field in a0 with defining polynomial x^14 + x^13 - 13*x^12 ˓→12*x^11 + 66*x^10 + 55*x^9 - 165*x^8 - 120*x^7 + 210*x^6 + 126*x^5 - 126*x^
˓→4 - 56*x^3 + 28*x^2 + 7*x - 1
To:
Cyclotomic Field of order 29 and degree 28
Defn: a0 |--> -a^27 - a^26 - a^25 - a^24 - a^23 - a^22 - a^21 - a^20 - a^19
˓→- a^18 - a^17 - a^16 - a^15 - a^14 - a^13 - a^12 - a^11 - a^10 - a^9 - a^8 ˓→ a^7 - a^6 - a^5 - a^4 - a^3 - a^2 - 1)
sage: F.<a> = NumberField(x^3 - 2)
sage: F.maximal_totally_real_subfield()
[Rational Field, Conversion map:
From: Rational Field
To:
Number Field in a with defining polynomial x^3 - 2]
sage: F.<a> = NumberField(x^4 - x^3 - x^2 + x + 1)
sage: F.maximal_totally_real_subfield()
[Rational Field, Conversion map:
From: Rational Field
To:
Number Field in a with defining polynomial x^4 - x^3 - x^2 + x + 1]
sage: F.<a> = NumberField(x^4 - x^3 + 2*x^2 + x + 1)
sage: F.maximal_totally_real_subfield()
[Number Field in a1 with defining polynomial x^2 - x - 1, Ring morphism:
From: Number Field in a1 with defining polynomial x^2 - x - 1
To:
Number Field in a with defining polynomial x^4 - x^3 + 2*x^2 + x + 1
Defn: a1 |--> -1/2*a^3 - 1/2]
sage: F.<a> = NumberField(x^4-4*x^2-x+1)
sage: F.maximal_totally_real_subfield()
[Number Field in a with defining polynomial x^4 - 4*x^2 - x + 1, Identity
˓→endomorphism of Number Field in a with defining polynomial x^4 - 4*x^2 - x
˓→+ 1]
An example of a relative extension where the base field is not the maximal totally real subfield.
sage: E_0.<a> = NumberField(x^2 - 4*x + 16)
sage: y = polygen(E_0)
sage: E.<z> = E_0.extension(y^2 - E_0.gen() / 2)
sage: E.maximal_totally_real_subfield()
[Number Field in z1 with defining polynomial x^2 - 2*x - 5, Composite map:
From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
To:
Number Field in z with defining polynomial x^2 - 1/2*a over its base
˓→field
Defn:
Ring morphism:
From: Number Field in z1 with defining polynomial x^2 - 2*x - 5
To:
Number Field in z with defining polynomial x^4 - 2*x^3 + x^2
˓→+ 6*x + 3
Defn: z1 |--> -1/3*z^3 + 1/3*z^2 + z - 1
then
Isomorphism map:
From: Number Field in z with defining polynomial x^4 - 2*x^3 + x^2
˓→+ 6*x + 3
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To:
Number Field in z with defining polynomial x^2 - 1/2*a over
its base field]
˓→
narrow_class_group ( proof=None)
Return the narrow class group of this field.
INPUT:
•proof - default: None (use the global proof setting, which defaults to True).
EXAMPLES:
sage: NumberField(x^3+x+9, 'a').narrow_class_group()
Multiplicative Abelian group isomorphic to C2
ngens ( )
Return the number of generators of this number field (always 1).
OUTPUT: the python integer 1.
EXAMPLES:
sage:
1
sage:
1
sage:
sage:
1
sage:
-3
NumberField(x^2 + 17,'a').ngens()
NumberField(x + 3,'a').ngens()
k.<a> = NumberField(x + 3)
k.ngens()
k.0
number_of_roots_of_unity ( )
Return the number of roots of unity in this field.
Note: We do not create the full unit group since that can be expensive, but we do use it if it is already
known.
EXAMPLES:
sage:
sage:
6
sage:
sage:
2
F.<alpha> = NumberField(x**22+3)
F.zeta_order()
F.<alpha> = NumberField(x**2-7)
F.zeta_order()
order ( )
Return the order of this number field (always +infinity).
OUTPUT: always positive infinity
EXAMPLES:
sage: NumberField(x^2 + 19,'a').order()
+Infinity
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pari_bnf ( proof=None, units=True)
PARI big number field corresponding to this field.
INPUT:
•proof – If False, assume GRH. If True, run PARI’s pari:bnfcertify to make sure that the results are
correct.
•units – (default: True) If True, insist on having fundamental units. If False, the units may or may
not be computed.
OUTPUT:
The PARI bnf structure of this number field.
Warning: Even with proof=True , I wouldn’t trust this to mean that everything computed involving
this number field is actually correct.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 1); k
Number Field in a with defining polynomial x^2 + 1
sage: len(k.pari_bnf())
10
sage: k.pari_bnf()[:4]
[[;], matrix(0,3), [;], ...]
sage: len(k.pari_nf())
9
sage: k.<a> = NumberField(x^7 + 7); k
Number Field in a with defining polynomial x^7 + 7
sage: dummy = k.pari_bnf(proof=True)
pari_nf ( important=True)
Return the PARI number field corresponding to this field.
INPUT:
•important – boolean (default: True ). If False , raise a RuntimeError if we need to do a
difficult discriminant factorization. This is useful when an integral basis is not strictly required, such
as for factoring polynomials over this number field.
OUTPUT:
The PARI number field obtained by calling
self.pari_polynomial('y') as argument.
the
PARI
function
pari:nfinit
with
Note: This method has the same effect as pari(self) .
EXAMPLES:
sage: k.<a> = NumberField(x^4 - 3*x + 7); k
Number Field in a with defining polynomial x^4 - 3*x + 7
sage: k.pari_nf()[:4]
[y^4 - 3*y + 7, [0, 2], 85621, 1]
sage: pari(k)[:4]
[y^4 - 3*y + 7, [0, 2], 85621, 1]
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sage: k.<a> = NumberField(x^4 - 3/2*x + 5/3); k
Number Field in a with defining polynomial x^4 - 3/2*x + 5/3
sage: k.pari_nf()
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ..., [1, y, 1/36*y^3 + 1/6*y^2 ˓→7, 1/6*y^2], [1, 0, 0, 252; 0, 1, 0, 0; 0, 0, 0, 36; 0, 0, 6, -36], [1, 0,
˓→0, 0, 0, 0, -18, 42, 0, -18, -46, -60, 0, 42, -60, -60; 0, 1, 0, 0, 1, 0,
˓→2, 0, 0, 2, -11, -1, 0, 0, -1, 9; 0, 0, 1, 0, 0, 0, 6, 6, 1, 6, -5, 0, 0,
˓→6, 0, 0; 0, 0, 0, 1, 0, 6, -6, -6, 0, -6, -1, 2, 1, -6, 2, 0]]
sage: pari(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]
sage: gp(k)
[y^4 - 324*y + 2160, [0, 2], 48918708, 216, ...]
With important=False , we simply bail out if we cannot easily factor the discriminant:
sage: p = next_prime(10^40); q = next_prime(10^41)
sage: K.<a> = NumberField(x^2 - p*q)
sage: K.pari_nf(important=False)
Traceback (most recent call last):
...
RuntimeError: Unable to factor discriminant with trial division
Next, we illustrate the maximize_at_primes and assume_disc_small parameters of
the NumberField constructor.
The following would take a very long time without the
maximize_at_primes option:
sage: K.<a> = NumberField(x^2 - p*q, maximize_at_primes=[p])
sage: K.pari_nf()
[y^2 - 100000000000000000000...]
Since the discriminant is square-free, this also works:
sage: K.<a> = NumberField(x^2 - p*q, assume_disc_small=True)
sage: K.pari_nf()
[y^2 - 100000000000000000000...]
pari_polynomial ( name=’x’)
Return the PARI polynomial corresponding to this number field.
INPUT:
•name – variable name (default: 'x' )
OUTPUT:
A monic polynomial with integral coefficients (PARI t_POL ) defining the PARI number field corresponding to self .
Warning: This is not the same as simply converting the defining polynomial to PARI.
EXAMPLES:
sage:
sage:
sage:
x^2 sage:
y = polygen(QQ)
k.<a> = NumberField(y^2 - 3/2*y + 5/3)
k.pari_polynomial()
x + 40
k.polynomial()._pari_()
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x^2 - 3/2*x + 5/3
sage: k.pari_polynomial('a')
a^2 - a + 40
Some examples with relative number fields:
sage:
sage:
x^4 +
sage:
a^4 +
sage:
x^4 +
sage:
x^2 +
k.<a, c> = NumberField([x^2 + 3, x^2 + 1])
k.pari_polynomial()
8*x^2 + 4
k.pari_polynomial('a')
8*a^2 + 4
k.absolute_polynomial()
8*x^2 + 4
k.relative_polynomial()
3
sage:
sage:
x^4 sage:
x^4 -
k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])
k.pari_polynomial()
x^2 + 1
k.absolute_polynomial()
x^2 + 1
This fails with arguments which are not a valid PARI variable name:
sage: k = QuadraticField(-1)
sage: k.pari_polynomial('I')
Traceback (most recent call last):
...
PariError: I already exists with incompatible valence
sage: k.pari_polynomial('i')
i^2 + 1
sage: k.pari_polynomial('theta')
Traceback (most recent call last):
...
PariError: theta already exists with incompatible valence
pari_rnfnorm_data ( L, proof=True)
Return the PARI pari:rnfisnorminit data corresponding to the extension L/self.
EXAMPLES:
sage:
sage:
sage:
sage:
8
x = polygen(QQ)
K = NumberField(x^2 - 2, 'alpha')
L = K.extension(x^2 + 5, 'gamma')
ls = K.pari_rnfnorm_data(L) ; len(ls)
sage:
sage:
sage:
sage:
8
K.<a> = NumberField(x^2 + x + 1)
P.<X> = K[]
L.<b> = NumberField(X^3 + a)
ls = K.pari_rnfnorm_data(L); len(ls)
pari_zk ( )
Integral basis of the PARI number field corresponding to this field.
This is the same as pari_nf().getattr(‘zk’), but much faster.
EXAMPLES:
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sage: k.<a> = NumberField(x^3 - 17)
sage: k.pari_zk()
[1, 1/3*y^2 - 1/3*y + 1/3, y]
sage: k.pari_nf().getattr('zk')
[1, 1/3*y^2 - 1/3*y + 1/3, y]
polynomial ( )
Return the defining polynomial of this number field.
This is exactly the same as self.defining_polynomial() .
EXAMPLES:
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial()
x^2 + 2/3*x - 9/17
polynomial_ntl ( )
Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.
This is used mainly to implement some internal arithmetic.
EXAMPLES:
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').polynomial_ntl()
([-27 34 51], 51)
polynomial_quotient_ring ( )
Return the polynomial quotient ring isomorphic to this number field.
EXAMPLES:
sage: K = NumberField(x^3 + 2*x - 5, 'alpha')
sage: K.polynomial_quotient_ring()
Univariate Quotient Polynomial Ring in alpha over Rational Field with modulus
˓→x^3 + 2*x - 5
polynomial_ring ( )
Return the polynomial ring that we view this number field as being a quotient of (by a principal ideal).
EXAMPLES: An example with an absolute field:
sage: k.<a> = NumberField(x^2 + 3)
sage: y = polygen(QQ, 'y')
sage: k.<a> = NumberField(y^2 + 3)
sage: k.polynomial_ring()
Univariate Polynomial Ring in y over Rational Field
An example with a relative field:
sage: y = polygen(QQ, 'y')
sage: M.<a> = NumberField([y^3 + 97, y^2 + 1]); M
Number Field in a0 with defining polynomial y^3 + 97 over its base field
sage: M.polynomial_ring()
Univariate Polynomial Ring in y over Number Field in a1 with defining
˓→polynomial y^2 + 1
power_basis ( )
Return a power basis for this number field over its base field.
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If this number field is represented as 𝑘[𝑡]/𝑓 (𝑡), then the basis returned is 1, 𝑡, 𝑡2 , . . . , 𝑡𝑑−1 where 𝑑 is the
degree of this number field over its base field.
EXAMPLES:
sage: K.<a> = NumberField(x^5 + 10*x + 1)
sage: K.power_basis()
[1, a, a^2, a^3, a^4]
sage: L.<b> = K.extension(x^2 - 2)
sage: L.power_basis()
[1, b]
sage: L.absolute_field('c').power_basis()
[1, c, c^2, c^3, c^4, c^5, c^6, c^7, c^8, c^9]
sage: M = CyclotomicField(15)
sage: M.power_basis()
[1, zeta15, zeta15^2, zeta15^3, zeta15^4, zeta15^5, zeta15^6, zeta15^7]
prime_above ( x, degree=None)
Return a prime ideal of self lying over x.
INPUT:
•x : usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes
x=0.
•degree (default: None): None or an integer. If one, find a prime above x of any degree. If an
integer, find a prime above x such that the resulting residue field has exactly this degree.
OUTPUT: A prime ideal of self lying over x. If degree is specified and no such ideal exists, raises a
ValueError.
EXAMPLES:
sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)
sage: P2 = F.prime_above(2)
sage: P2 # random
Fractional ideal (-t)
sage: 2 in P2
True
sage: P2.is_prime()
True
sage: P2.norm()
2
sage: P3 = F.prime_above(3)
sage: P3 # random
Fractional ideal (t + 1)
sage: 3 in P3
True
sage: P3.is_prime()
True
sage: P3.norm()
3
The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:
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sage: F.prime_above(3, degree=2)
Traceback (most recent call last):
...
ValueError: No prime of degree 2 above Fractional ideal (3)
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]
Asking for a specific degree works:
sage: P5_1 = F.prime_above(5, degree=1)
sage: P5_1 # random
Fractional ideal (-t^2 - 1)
sage: P5_1.residue_class_degree()
1
sage: P5_2 = F.prime_above(5, degree=2)
sage: P5_2 # random
Fractional ideal (t^2 - 2*t - 1)
sage: P5_2.residue_class_degree()
2
Relative number fields are ok:
sage: G = F.extension(x^2 - 11, 'b')
sage: G.prime_above(7)
Fractional ideal (b + 2)
It doesn’t make sense to factor the ideal (0):
sage: F.prime_above(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'
prime_factors ( x)
Return a list of the prime ideals of self which divide the ideal generated by 𝑥.
OUTPUT: list of prime ideals (a new list is returned each time this function is called)
EXAMPLES:
sage: K.<w> = NumberField(x^2 + 23)
sage: K.prime_factors(w + 1)
[Fractional ideal (2, 1/2*w - 1/2), Fractional ideal (2, 1/2*w + 1/2),
˓→Fractional ideal (3, 1/2*w + 1/2)]
primes_above ( x, degree=None)
Return prime ideals of self lying over x.
INPUT:
•x : usually an element or ideal of self. It should be such that self.ideal(x) is sensible. This excludes
x=0.
•degree (default: None): None or an integer. If None, find all primes above x of any degree. If an
integer, find all primes above x such that the resulting residue field has exactly this degree.
OUTPUT: A list of prime ideals of self lying over x. If degree is specified and no such ideal exists, returns
the empty list. The output is sorted by residue degree first, then by underlying prime (or equivalently, by
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norm).
EXAMPLES:
sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)
sage: P2s = F.primes_above(2)
sage: P2s # random
[Fractional ideal (-t)]
sage: all(2 in P2 for P2 in P2s)
True
sage: all(P2.is_prime() for P2 in P2s)
True
sage: [ P2.norm() for P2 in P2s ]
[2]
sage: P3s = F.primes_above(3)
sage: P3s # random
[Fractional ideal (t + 1)]
sage: all(3 in P3 for P3 in P3s)
True
sage: all(P3.is_prime() for P3 in P3s)
True
sage: [ P3.norm() for P3 in P3s ]
[3]
The ideal (3) is totally ramified in F, so there is no degree 2 prime above 3:
sage: F.primes_above(3, degree=2)
[]
sage: [ id.residue_class_degree() for id, _ in F.ideal(3).factor() ]
[1]
Asking for a specific degree works:
sage: P5_1s = F.primes_above(5, degree=1)
sage: P5_1s # random
[Fractional ideal (-t^2 - 1)]
sage: P5_1 = P5_1s[0]; P5_1.residue_class_degree()
1
sage: P5_2s = F.primes_above(5, degree=2)
sage: P5_2s # random
[Fractional ideal (t^2 - 2*t - 1)]
sage: P5_2 = P5_2s[0]; P5_2.residue_class_degree()
2
Works in relative extensions too:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
True
64
PQ.<X> = QQ[]
F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = F.ideal(a + 2*b)
P, Q = K.primes_above(I)
K.ideal(I) == P^4*Q
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sage: K.primes_above(I, degree=1) == [P]
True
sage: K.primes_above(I, degree=4) == [Q]
True
It doesn’t make sense to factor the ideal (0), so this raises an error:
sage: F.prime_above(0)
Traceback (most recent call last):
...
AttributeError: 'NumberFieldIdeal' object has no attribute 'prime_factors'
primes_of_bounded_norm ( B)
Returns a sorted list of all prime ideals with norm at most 𝐵.
INPUT:
•B – a positive integer; upper bound on the norms of the primes generated.
OUTPUT:
A list of all prime ideals of this number field of norm at most 𝐵, sorted by norm. Primes of the same norm
are sorted using the comparison function for ideals, which is based on the Hermite Normal Form.
Note: See also primes_of_bounded_norm_iter() for an iterator version of this, but note that
the iterator sorts the primes in order of underlying rational prime, not by norm.
EXAMPLES:
sage: K.<i> = QuadraticField(-1)
sage: K.primes_of_bounded_norm(10)
[Fractional ideal (i + 1), Fractional ideal (-i - 2), Fractional ideal (2*i +
˓→1), Fractional ideal (3)]
sage: K.primes_of_bounded_norm(1)
[]
sage: K.<a> = NumberField(x^3-2)
sage: P = K.primes_of_bounded_norm(30)
sage: P
[Fractional ideal (a),
Fractional ideal (a + 1),
Fractional ideal (-a^2 - 1),
Fractional ideal (a^2 + a - 1),
Fractional ideal (2*a + 1),
Fractional ideal (-2*a^2 - a - 1),
Fractional ideal (a^2 - 2*a - 1),
Fractional ideal (a + 3)]
sage: [p.norm() for p in P]
[2, 3, 5, 11, 17, 23, 25, 29]
primes_of_bounded_norm_iter ( B)
Iterator yielding all prime ideals with norm at most 𝐵.
INPUT:
•B – a positive integer; upper bound on the norms of the primes generated.
OUTPUT:
An iterator over all prime ideals of this number field of norm at most 𝐵.
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Note: The output is not sorted by norm, but by size of the underlying rational prime.
EXAMPLES:
sage: K.<i> = QuadraticField(-1)
sage: it = K.primes_of_bounded_norm_iter(10)
sage: list(it)
[Fractional ideal (i + 1),
Fractional ideal (3),
Fractional ideal (-i - 2),
Fractional ideal (2*i + 1)]
sage: list(K.primes_of_bounded_norm_iter(1))
[]
primes_of_degree_one_iter ( num_integer_primes=10000, max_iterations=100)
Return an iterator yielding prime ideals of absolute degree one and small norm.
Warning: It is possible that there are no primes of 𝐾 of absolute degree one of small prime norm,
and it possible that this algorithm will not find any primes of small norm.
See module sage.rings.number_field.small_primes_of_degree_one for details.
INPUT:
•num_integer_primes (default: 10000) - an integer. We try to find primes of absolute
norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes
is 2, the largest norm found will be 3, since the second prime is 3.
•max_iterations (default: 100) - an integer. We test max_iterations integers to find
small primes before raising StopIteration.
EXAMPLES:
sage: K.<z> = CyclotomicField(10)
sage: it = K.primes_of_degree_one_iter()
sage: Ps = [ next(it) for i in range(3) ]
sage: Ps # random
[Fractional ideal (z^3 + z + 1), Fractional ideal (3*z^3 - z^2 + z - 1),
˓→Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [ P.norm() for P in Ps ] # random
[11, 31, 41]
sage: [ P.residue_class_degree() for P in Ps ]
[1, 1, 1]
primes_of_degree_one_list ( n, num_integer_primes=10000, max_iterations=100)
Return a list of n prime ideals of absolute degree one and small norm.
Warning: It is possible that there are no primes of 𝐾 of absolute degree one of small prime norm,
and it possible that this algorithm will not find any primes of small norm.
See module sage.rings.number_field.small_primes_of_degree_one for details.
INPUT:
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•num_integer_primes (default: 10000) - an integer. We try to find primes of absolute
norm no greater than the num_integer_primes-th prime number. For example, if num_integer_primes
is 2, the largest norm found will be 3, since the second prime is 3.
•max_iterations (default: 100) - an integer. We test max_iterations integers to find
small primes before raising StopIteration.
EXAMPLES:
sage: K.<z> = CyclotomicField(10)
sage: Ps = K.primes_of_degree_one_list(3)
sage: Ps # random output
[Fractional ideal (-z^3 - z^2 + 1), Fractional ideal (2*z^3 - 2*z^2 + 2*z ˓→3), Fractional ideal (2*z^3 - 3*z^2 + z - 2)]
sage: [ P.norm() for P in Ps ]
[11, 31, 41]
sage: [ P.residue_class_degree() for P in Ps ]
[1, 1, 1]
primitive_element ( )
Return a primitive element for this field, i.e., an element that generates it over Q.
EXAMPLES:
sage:
sage:
a
sage:
sage:
a - b
sage:
a - b
sage:
x^2 +
sage:
x^8 -
K.<a> = NumberField(x^3 + 2)
K.primitive_element()
K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
K.primitive_element()
+ c
alpha = K.primitive_element(); alpha
+ c
alpha.minpoly()
(2*b - 2*c)*x - 2*c*b + 6
alpha.absolute_minpoly()
40*x^6 + 352*x^4 - 960*x^2 + 576
primitive_root_of_unity ( )
Return a generator of the roots of unity in this field.
OUTPUT: a primitive root of unity. No guarantee is made about which primitive root of unity this returns,
not even for cyclotomic fields. Repeated calls of this function may return a different value.
Note: We do not create the full unit group since that can be expensive, but we do use it if it is already
known.
EXAMPLES:
sage: K.<i> = NumberField(x^2+1)
sage: z = K.primitive_root_of_unity(); z
i
sage: z.multiplicative_order()
4
sage: K.<a> = NumberField(x^2+x+1)
sage: z = K.primitive_root_of_unity(); z
a + 1
sage: z.multiplicative_order()
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6
sage:
sage:
sage:
sage:
sage:
-1
x = polygen(QQ)
F.<a,b> = NumberField([x^2 - 2, x^2 - 3])
y = polygen(F)
K.<c> = F.extension(y^2 - (1 + a)*(a + b)*a*b)
K.primitive_root_of_unity()
We do not special-case cyclotomic fields, so we do not always get the most obvious primitive root of unity:
sage: K.<a> = CyclotomicField(3)
sage: z = K.primitive_root_of_unity(); z
a + 1
sage: z.multiplicative_order()
6
sage:
sage:
zeta3
sage:
6
K = CyclotomicField(3)
z = K.primitive_root_of_unity(); z
+ 1
z.multiplicative_order()
random_element ( num_bound=None, den_bound=None, integral_coefficients=False, distribution=None)
Return a random element of this number field.
INPUT:
•num_bound - Bound on numerator of the coefficients of the resulting element
•den_bound - Bound on denominators of the coefficients of the resulting element
•integral_coefficients (default: False) - If True, then the resulting element will have integral coefficients. This option overrides any value of 𝑑𝑒𝑛𝑏 𝑜𝑢𝑛𝑑.
•distribution - Distribution to use for the coefficients of the resulting element
OUTPUT:
•Element of this number field
EXAMPLES:
sage: K.<j> = NumberField(x^8+1)
sage: K.random_element()
1/2*j^7 - j^6 - 12*j^5 + 1/2*j^4 - 1/95*j^3 - 1/2*j^2 - 4
sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
sage: K.random_element()
((6136*c - 7489/3)*b + 5825/3*c - 71422/3)*a + (-4849/3*c + 58918/3)*b ˓→45718/3*c + 75409/12
sage: K.<a> = NumberField(x^5-2)
sage: K.random_element(integral_coefficients=True)
a^3 + a^2 - 3*a - 1
real_embeddings ( prec=53)
Return all homomorphisms of this number field into the approximate real field with precision prec.
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If prec is 53 (the default), then the real double field is used; otherwise the arbitrary precision (but
slow) real field is used. If you want embeddings into the 53-bit double precision, which is faster, use
self.embeddings(RDF) .
Note: This function uses finite precision real numbers. In functions that should output proven results, one
could use self.embeddings(AA) instead.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 2)
sage: K.real_embeddings()
[
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Real Field with 53 bits of precision
Defn: a |--> -1.25992104989487
]
sage: K.real_embeddings(16)
[
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Real Field with 16 bits of precision
Defn: a |--> -1.260
]
sage: K.real_embeddings(100)
[
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Real Field with 100 bits of precision
Defn: a |--> -1.2599210498948731647672106073
]
As this is a numerical function, the number of embeddings may be incorrect if the precision is too low:
sage:
sage:
2
sage:
2
sage:
0
sage:
0
K = NumberField(x^2+2*10^1000*x + 10^2000+1, 'a')
len(K.real_embeddings())
len(K.real_embeddings(100))
len(K.real_embeddings(10000))
len(K.embeddings(AA))
reduced_basis ( prec=None)
This function returns an LLL-reduced basis for the Minkowski-embedding of the maximal order of a
number field.
INPUT:
•self - number field, the base field
•prec (default:
None) - the precision with which to compute the Minkowski embedding.
OUTPUT:
An LLL-reduced basis for the Minkowski-embedding of the maximal order of a number field, given by a
sequence of (integral) elements from the field.
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Note: In the non-totally-real case, the LLL routine we call is currently PARI’s pari:qflll, which works
with floating point approximations, and so the result is only as good as the precision promised by PARI.
The matrix returned will always be integral; however, it may only be only “almost” LLL-reduced when
the precision is not sufficiently high.
EXAMPLES:
sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)
sage: F.maximal_order().basis()
[1/2*t^5 + 1/2*t^4 + 1/2*t^2 + 1/2, t, t^2, t^3, t^4, t^5]
sage: F.reduced_basis()
[-1, -1/2*t^5 + 1/2*t^4 + 3*t^3 - 3/2*t^2 - 4*t - 1/2, t, 1/2*t^5 + 1/2*t^4 ˓→4*t^3 - 5/2*t^2 + 7*t + 1/2, 1/2*t^5 - 1/2*t^4 - 2*t^3 + 3/2*t^2 - 1/2, 1/
˓→2*t^5 - 1/2*t^4 - 3*t^3 + 5/2*t^2 + 4*t - 5/2]
sage: CyclotomicField(12).reduced_basis()
[1, zeta12^2, zeta12, zeta12^3]
reduced_gram_matrix ( prec=None)
This function returns the Gram matrix of an LLL-reduced basis for the Minkowski embedding of the
maximal order of a number field.
INPUT:
•self - number field, the base field
•prec (default:
(See NOTE below.)
None) - the precision with which to calculate the Minkowski embedding.
OUTPUT: The Gram matrix [⟨𝑥𝑖 , 𝑥𝑗 ⟩] of an LLL reduced basis for the maximal order of self,
where the integral basis for self is given by {𝑥0 , . . . , 𝑥𝑛−1 }. Here ⟨, ⟩ is the usual inner product on R𝑛 , and self is embedded in R𝑛 by the Minkowski embedding. See the docstring for
NumberField_absolute.Minkowski_embedding() for more information.
Note: In the non-totally-real case, the LLL routine we call is currently PARI’s pari:qflll, which works
with floating point approximations, and so the result is only as good as the precision promised by PARI.
In particular, in this case, the returned matrix will not be integral, and may not have enough precision to
recover the correct gram matrix (which is known to be integral for theoretical reasons). Thus the need for
the prec flag above.
If the following run-time error occurs: “PariError: not a definite matrix in lllgram (42)” try increasing the
prec parameter,
EXAMPLES:
sage: F.<t> = NumberField(x^6-7*x^4-x^3+11*x^2+x-1)
sage: F.reduced_gram_matrix()
[ 6 3 0 2 0 1]
[ 3 9 0 1 0 -2]
[ 0 0 14 6 -2 3]
[ 2 1 6 16 -3 3]
[ 0 0 -2 -3 16 6]
[ 1 -2 3 3 6 19]
sage: Matrix(6, [(x*y).trace() for x in F.integral_basis() for y in F.
˓→integral_basis()])
[2550 133 259 664 1368 3421]
[ 133
14
3
54
30 233]
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[ 259
[ 664
[1368
[3421
3
54
30
233
54
30 233 217]
30 233 217 1078]
233 217 1078 1371]
217 1078 1371 5224]
sage: var('x')
x
sage: F.<alpha> = NumberField(x^4+x^2+712312*x+131001238)
sage: F.reduced_gram_matrix(prec=128)
[
4.0000000000000000000000000000000000000
0.
˓→00000000000000000000000000000000000000 -2.
˓→1369360000000000000000000000000000000e6 -3.
˓→3122478000000000000000000000000000000e7]
[ 0.00000000000000000000000000000000000000
46721.
˓→539331563218381658483353092335550 -2.
˓→2467769057394530109094755223395819322e7 -3.
˓→4807276041138450473611629088647496430e8]
[-2.1369360000000000000000000000000000000e6 -2.
˓→2467769057394530109094755223395819322e7 7.
˓→0704285924714907491782135494859351061e12 1.
˓→1256639928034037006027526953641297995e14]
[-3.3122478000000000000000000000000000000e7 -3.
˓→4807276041138450473611629088647496430e8 1.
˓→1256639928034037006027526953641297995e14 1.
˓→7923838231014970520503146603069479547e15]
regulator ( proof=None)
Return the regulator of this number field.
Note that PARI computes the regulator to higher precision than the Sage default.
INPUT:
•proof - default: True, unless you set it otherwise.
EXAMPLES:
sage: NumberField(x^2-2, 'a').regulator()
0.881373587019543
sage: NumberField(x^4+x^3+x^2+x+1, 'a').regulator()
0.962423650119207
residue_field ( prime, names=None, check=True)
Return the residue field of this number field at a given prime, ie 𝑂𝐾 /𝑝𝑂𝐾 .
INPUT:
•prime - a prime ideal of the maximal order in this number field, or an element of the field which
generates a principal prime ideal.
•names - the name of the variable in the residue field
•check - whether or not to check the primality of prime.
OUTPUT: The residue field at this prime.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: P = K.ideal(61).factor()[0][0]
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sage: K.residue_field(P)
Residue field in abar of Fractional ideal (61, a^2 + 30)
sage: K.<i> = NumberField(x^2 + 1)
sage: K.residue_field(1+i)
Residue field of Fractional ideal (i + 1)
sage: L.residue_field(L.prime_above(5)^2)
Traceback (most recent call last):
...
ValueError: Fractional ideal (5) is not a prime ideal
roots_of_unity ( )
Return all the roots of unity in this field, primitive or not.
EXAMPLES:
sage: K.<b> = NumberField(x^2+1)
sage: zs = K.roots_of_unity(); zs
[b, -1, -b, 1]
sage: [ z**K.number_of_roots_of_unity() for z in zs ]
[1, 1, 1, 1]
selmer_group ( S, m, proof=True, orders=False)
Compute the group 𝐾(𝑆, 𝑚).
INPUT:
•S – a set of primes of self
•m – a positive integer
•proof – if False, assume the GRH in computing the class group
•orders (default False) – if True, output two lists, the generators and their orders
OUTPUT:
A list of generators of 𝐾(𝑆, 𝑚), and (optionally) their orders as elements of 𝐾 × /(𝐾 × )𝑚 . This is the
subgroup of 𝐾 × /(𝐾 × )𝑚 consisting of elements 𝑎 such that the valuation of 𝑎 is divisible by 𝑚 at all
primes not in 𝑆. It fits in an exact sequence between the units modulo 𝑚-th powers and the 𝑚-torsion in
the 𝑆-class group:
×
×
1 −→ 𝑂𝐾,𝑆
/(𝑂𝐾,𝑆
)𝑚 −→ 𝐾(𝑆, 𝑚) −→ Cl𝐾,𝑆 [𝑚] −→ 0.
√
The group 𝐾(𝑆, 𝑚) contains the subgroup of those 𝑎 such that 𝐾( 𝑚 𝑎)/𝐾 is unramified at all primes of
𝐾 outside of 𝑆, but may contain it properly when not all primes dividing 𝑚 are in 𝑆.
EXAMPLES:
sage: K.<a> = QuadraticField(-5)
sage: K.selmer_group((), 2)
[-1, 2]
The previous example shows that the group generated by the output may be strictly larger than the ‘true’
Selmer group of elements giving extensions unramified outside 𝑆, since that has order just 2, generated by
−1:
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sage: K.class_number()
2
sage: K.hilbert_class_field('b')
Number Field in b with defining polynomial x^2 + 1 over its base field
When 𝑚 is prime all the orders are equal to 𝑚, but in general they are only divisors of 𝑚:
sage: K.<a> = QuadraticField(-5)
sage: P2 = K.ideal(2, -a+1)
sage: P3 = K.ideal(3, a+1)
sage: K.selmer_group((), 2, orders=True)
([-1, 2], [2, 2])
sage: K.selmer_group((), 4, orders=True)
([-1, 4], [2, 2])
sage: K.selmer_group([P2], 2)
[2, -1]
sage: K.selmer_group((P2,P3), 4)
[2, -a - 1, -1]
sage: K.selmer_group((P2,P3), 4, orders=True)
([2, -a - 1, -1], [4, 4, 2])
sage: K.selmer_group([P2], 3)
[2]
sage: K.selmer_group([P2, P3], 3)
[2, -a - 1]
sage: K.selmer_group([P2, P3, K.ideal(a)], 3)
[2, a + 1, a]
# random signs
Example over Q (as a number field):
sage: K.<a> = NumberField(polygen(QQ))
sage: K.selmer_group([],5)
[]
sage: K.selmer_group([K.prime_above(p) for p in [2,3,5]],2)
[2, 3, 5, -1]
sage: K.selmer_group([K.prime_above(p) for p in [2,3,5]],6, orders=True)
([2, 3, 5, -1], [6, 6, 6, 2])
selmer_group_iterator ( S, m, proof=True)
Return an iterator through elements of the finite group 𝐾(𝑆, 𝑚).
INPUT:
•S – a set of primes of self
•m – a positive integer
•proof – if False, assume the GRH in computing the class group
OUTPUT:
An iterator yielding the distinct elements of 𝐾(𝑆, 𝑚).
NumberField_generic.selmer_group() for more information.
See
the
docstring
for
EXAMPLES:
sage: K.<a> = QuadraticField(-5)
sage: list(K.selmer_group_iterator((), 2))
[1, 2, -1, -2]
sage: list(K.selmer_group_iterator((), 4))
[1, 4, -1, -4]
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sage: list(K.selmer_group_iterator([K.ideal(2, -a+1)], 2))
[1, -1, 2, -2]
sage: list(K.selmer_group_iterator([K.ideal(2, -a+1), K.ideal(3, a+1)], 2))
[1, -1, -a - 1, a + 1, 2, -2, -2*a - 2, 2*a + 2]
Examples over Q (as a number field):
sage: K.<a> = NumberField(polygen(QQ))
sage: list(K.selmer_group_iterator([], 5))
[1]
sage: list(K.selmer_group_iterator([], 4))
[1, -1]
sage: list(K.selmer_group_iterator([K.prime_above(p) for p in [11,13]],2))
[1, -1, 13, -13, 11, -11, 143, -143]
signature ( )
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of
this field, respectively.
EXAMPLES:
sage: NumberField(x^2+1, 'a').signature()
(0, 1)
sage: NumberField(x^3-2, 'a').signature()
(1, 1)
solve_CRT ( reslist, Ilist, check=True)
Solve a Chinese remainder problem over this number field.
INPUT:
•reslist – a list of residues, i.e. integral number field elements
•Ilist – a list of integral ideals, assumed pairsise coprime
•check (boolean, default True) – if True, result is checked
OUTPUT:
An integral element x such that x-reslist[i] is in Ilist[i] for all i.
Note: The current implementation requires the ideals to be pairwise coprime. A more general version
would be possible.
EXAMPLES:
sage: K.<a> = NumberField(x^2-10)
sage: Ilist = [K.primes_above(p)[0] for p in prime_range(10)]
sage: b = K.solve_CRT([1,2,3,4],Ilist,True)
sage: all([b-i-1 in Ilist[i] for i in range(4)])
True
sage: Ilist = [K.ideal(a), K.ideal(2)]
sage: K.solve_CRT([0,1],Ilist,True)
Traceback (most recent call last):
...
ArithmeticError: ideals in solve_CRT() must be pairwise coprime
sage: Ilist[0]+Ilist[1]
Fractional ideal (2, a)
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specified_complex_embedding ( )
Returns the embedding of this field into the complex numbers which has been specified.
Fields created with the QuadraticField or CyclotomicField constructors come with an implicit
embedding. To get one of these fields without the embedding, use the generic NumberField constructor.
EXAMPLES:
sage: QuadraticField(-1, 'I').specified_complex_embedding()
Generic morphism:
From: Number Field in I with defining polynomial x^2 + 1
To:
Complex Lazy Field
Defn: I -> 1*I
sage: QuadraticField(3, 'a').specified_complex_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^2 - 3
To:
Real Lazy Field
Defn: a -> 1.732050807568878?
sage: CyclotomicField(13).specified_complex_embedding()
Generic morphism:
From: Cyclotomic Field of order 13 and degree 12
To:
Complex Lazy Field
Defn: zeta13 -> 0.885456025653210? + 0.464723172043769?*I
Most fields don’t implicitly have embeddings unless explicitly specified:
sage: NumberField(x^2-2, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3-x+5, 'a').specified_complex_embedding() is None
True
sage: NumberField(x^3-x+5, 'a', embedding=2).specified_complex_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 5
To:
Real Lazy Field
Defn: a -> -1.904160859134921?
sage: NumberField(x^3-x+5, 'a', embedding=CDF.0).specified_complex_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 5
To:
Complex Lazy Field
Defn: a -> 0.952080429567461? + 1.311248044077123?*I
This function only returns complex embeddings:
sage: K.<a> = NumberField(x^2-2, embedding=Qp(7)(2).sqrt())
sage: K.specified_complex_embedding() is None
True
sage: K.gen_embedding()
3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^9 + 6*7^10 +
˓→2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19 + O(7^20)
sage: K.coerce_embedding()
Generic morphism:
From: Number Field in a with defining polynomial x^2 - 2
To:
7-adic Field with capped relative precision 20
Defn: a -> 3 + 7 + 2*7^2 + 6*7^3 + 7^4 + 2*7^5 + 7^6 + 2*7^7 + 4*7^8 + 6*7^
˓→9 + 6*7^10 + 2*7^11 + 7^12 + 7^13 + 2*7^15 + 7^16 + 7^17 + 4*7^18 + 6*7^19
˓→+ O(7^20)
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structure ( )
Return fixed isomorphism or embedding structure on self.
This is used to record various isomorphisms or embeddings that arise naturally in other constructions.
EXAMPLES:
sage: K.<z> = NumberField(x^2 + 3)
sage: L.<a> = K.absolute_field(); L
Number Field in a with defining polynomial x^2 + 3
sage: L.structure()
(Isomorphism given by variable name change map:
From: Number Field in a with defining polynomial
To:
Number Field in z with defining polynomial
Isomorphism given by variable name change map:
From: Number Field in z with defining polynomial
To:
Number Field in a with defining polynomial
sage:
sage:
sage:
sage:
sage:
-a
x^2 + 3
x^2 + 3,
x^2 + 3
x^2 + 3)
K.<a> = QuadraticField(-3)
R.<y> = K[]
D.<x0> = K.extension(y)
D_abs.<y0> = D.absolute_field()
D_abs.structure()[0](y0)
subfield ( alpha, name=None, names=None)
Return a number field 𝐾 isomorphic to Q(𝛼) (if this is an absolute number field) or 𝐿(𝛼) (if this is a
relative extension 𝑀/𝐿) and a map from K to self that sends the generator of K to alpha.
INPUT:
•alpha - an element of self, or something that coerces to an element of self.
OUTPUT:
•K - a number field
•from_K - a homomorphism from K to self that sends the generator of K to alpha.
EXAMPLES:
sage: K.<a> = NumberField(x^4 - 3); K
Number Field in a with defining polynomial x^4 - 3
sage: H.<b>, from_H = K.subfield(a^2)
sage: H
Number Field in b with defining polynomial x^2 - 3
sage: from_H(b)
a^2
sage: from_H
Ring morphism:
From: Number Field in b with defining polynomial x^2 - 3
To:
Number Field in a with defining polynomial x^4 - 3
Defn: b |--> a^2
A relative example.
Note that the result returned is the subfield generated by 𝛼 over
self.base_field() , not over Q (see trac ticket #5392):
sage: L.<a> = NumberField(x^2 - 3)
sage: M.<b> = L.extension(x^4 + 1)
sage: K, phi = M.subfield(b^2)
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sage: K.base_field() is L
True
Subfields inherit embeddings:
sage: K.<z> = CyclotomicField(5)
sage: L, K_from_L = K.subfield(z-z^2-z^3+z^4)
sage: L
Number Field in z0 with defining polynomial x^2 - 5
sage: CLF_from_K = K.coerce_embedding(); CLF_from_K
Generic morphism:
From: Cyclotomic Field of order 5 and degree 4
To:
Complex Lazy Field
Defn: z -> 0.309016994374948? + 0.951056516295154?*I
sage: CLF_from_L = L.coerce_embedding(); CLF_from_L
Generic morphism:
From: Number Field in z0 with defining polynomial x^2 - 5
To:
Complex Lazy Field
Defn: z0 -> 2.236067977499790?
Check transitivity:
sage: CLF_from_L(L.gen())
2.236067977499790?
sage: CLF_from_K(K_from_L(L.gen()))
2.23606797749979? + 0.?e-14*I
If 𝑠𝑒𝑙𝑓 has no specified embedding, then 𝐾 comes with an embedding in 𝑠𝑒𝑙𝑓 :
sage: K.<a> = NumberField(x^6 - 6*x^4 + 8*x^2 - 1)
sage: L.<b>, from_L = K.subfield(a^2)
sage: L
Number Field in b with defining polynomial x^3 - 6*x^2 + 8*x - 1
sage: L.gen_embedding()
a^2
You can also view a number field as having a different generator by just choosing the input to generate the
whole field; for that it is better to use self.change_generator , which gives isomorphisms in both
directions.
trace_dual_basis ( b)
Compute the dual basis of a basis of self with respect to the trace pairing.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + x + 1)
sage: b = [1, 2*a, 3*a^2]
sage: T = K.trace_dual_basis(b); T
[4/31*a^2 - 6/31*a + 13/31, -9/62*a^2 - 1/31*a - 3/31, 2/31*a^2 - 3/31*a + 4/
˓→93]
sage: [(b[i]*T[j]).trace() for i in range(3) for j in range(3)]
[1, 0, 0, 0, 1, 0, 0, 0, 1]
trace_pairing ( v)
Return the matrix of the trace pairing on the elements of the list 𝑣.
EXAMPLES:
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sage: K.<zeta3> = NumberField(x^2 + 3)
sage: K.trace_pairing([1,zeta3])
[ 2 0]
[ 0 -6]
uniformizer ( P, others=’positive’)
Returns an element of self with valuation 1 at the prime ideal P.
INPUT:
•self - a number field
•P - a prime ideal of self
•others - either “positive” (default), in which case the element will have non-negative valuation at
all other primes of self, or “negative”, in which case the element will have non-positive valuation at
all other primes of self.
Note: When P is principal (e.g. always when self has class number one) the result may or may not be a
generator of P!
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 5); K
Number Field in a with defining polynomial x^2 + 5
sage: P,Q = K.ideal(3).prime_factors()
sage: P
Fractional ideal (3, a + 1)
sage: pi = K.uniformizer(P); pi
a + 1
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1)) * (Fractional ideal (3, a + 1))
sage: pi = K.uniformizer(P,'negative'); pi
1/2*a + 1/2
sage: K.ideal(pi).factor()
(Fractional ideal (2, a + 1))^-1 * (Fractional ideal (3, a + 1))
sage: K = CyclotomicField(9)
sage: Plist=K.ideal(17).prime_factors()
sage: pilist = [K.uniformizer(P) for P in Plist]
sage: [pi.is_integral() for pi in pilist]
[True, True, True]
sage: [pi.valuation(P) for pi,P in zip(pilist,Plist)]
[1, 1, 1]
sage: [ pilist[i] in Plist[i] for i in range(len(Plist)) ]
[True, True, True]
sage: K.<t> = NumberField(x^4 sage: [K.uniformizer(P) for P,e
[2]
sage: [K.uniformizer(P) for P,e
[t - 1]
sage: [K.uniformizer(P) for P,e
[t^2 - t + 1, t + 2, t - 2]
sage: [K.uniformizer(P) for P,e
[t^2 + 3*t + 1]
78
x^3 - 3*x^2 - x + 1)
in factor(K.ideal(2))]
in factor(K.ideal(3))]
in factor(K.ideal(5))]
in factor(K.ideal(7))]
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sage: [K.uniformizer(P) for P,e in factor(K.ideal(67))]
[t + 23, t + 26, t - 32, t - 18]
ALGORITHM:
Use PARI. More precisely, use the second component of pari:idealprimedec in the “positive”
case. Use pari:idealappr with exponent of -1 and invert the result in the “negative” case.
unit_group ( proof=None)
Return the unit group (including torsion) of this number field.
ALGORITHM: Uses PARI’s pari:bnfunit command.
INPUT:
•proof (bool, default True) flag passed to pari .
Note: The group is cached.
See also:
units() S_unit_group() S_units()
EXAMPLES:
sage: x = QQ['x'].0
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: U = K.unit_group(); U
Unit group with structure C10 x Z of Number Field in a with defining
˓→polynomial x^4 - 10*x^3 + 100*x^2 - 375*x + 1375
sage: U.gens()
(u0, u1)
sage: U.gens_values() # random
[-1/275*a^3 + 7/55*a^2 - 6/11*a + 4, 1/275*a^3 + 4/55*a^2 - 5/11*a + 3]
sage: U.invariants()
(10, 0)
sage: [u.multiplicative_order() for u in U.gens()]
[10, +Infinity]
For big number fields, provably computing the unit group can take a very long time. In this case, one can
ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):
sage: K = NumberField(x^17 + 3, 'a')
sage: K.unit_group(proof=True) # takes forever, not tested
...
sage: U = K.unit_group(proof=False)
sage: U
Unit group with structure C2 x Z x Z x Z x Z x Z x Z x Z x Z of Number Field
˓→in a with defining polynomial x^17 + 3
sage: U.gens()
(u0, u1, u2, u3, u4, u5, u6, u7, u8)
sage: U.gens_values() # result not independently verified
[-1, a^9 + a - 1, a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^
˓→4 + 4*a^3 - 3*a^2 - 2*a + 2, a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8
˓→+ a^7 - 2*a^6 + a^4 - 3*a^3 + 2*a^2 - 2*a + 1, 2*a^16 - a^14 - a^13 + 3*a^
˓→12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4 - 2*a^3 - 2*a^2 + 3*a +
˓→4, 2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 ˓→ 5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4, a^16 - a^15 - 3*a^14 ˓→4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^7 + 4*a^6 + 2*a^5 ˓→2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7, a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 +
˓→2*a^7 - a^5 + 2*a^3 - a^2 - 3*a + 1, 5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^
1.1. Number
Fields
79
˓→11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 - 7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^
˓→2 + 13*a + 4]
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units ( proof=None)
Return generators for the unit group modulo torsion.
ALGORITHM: Uses PARI’s pari:bnfunit command.
INPUT:
•proof (bool, default True) flag passed to pari .
Note: For more functionality see the unit_group() function.
See also:
unit_group() S_unit_group() S_units()
EXAMPLES:
sage: x = polygen(QQ)
sage: A = x^4 - 10*x^3 + 20*5*x^2 - 15*5^2*x + 11*5^3
sage: K = NumberField(A, 'a')
sage: K.units()
(1/275*a^3 + 4/55*a^2 - 5/11*a + 3,)
For big number fields, provably computing the unit group can take a very long time. In this case, one can
ask for the conjectural unit group (correct if the Generalized Riemann Hypothesis is true):
sage: K = NumberField(x^17 + 3, 'a')
sage: K.units(proof=True) # takes forever, not tested
...
sage: K.units(proof=False) # result not independently verified
(a^9 + a - 1,
a^15 - a^12 + a^10 - a^9 - 2*a^8 + 3*a^7 + a^6 - 3*a^5 + a^4 + 4*a^3 - 3*a^2
˓→- 2*a + 2,
a^16 - a^15 + a^14 - a^12 + a^11 - a^10 - a^8 + a^7 - 2*a^6 + a^4 - 3*a^3 +
˓→2*a^2 - 2*a + 1,
2*a^16 - a^14 - a^13 + 3*a^12 - 2*a^10 + a^9 + 3*a^8 - 3*a^6 + 3*a^5 + 3*a^4
˓→- 2*a^3 - 2*a^2 + 3*a + 4,
2*a^16 - 3*a^15 + 3*a^14 - 3*a^13 + 3*a^12 - a^11 + a^9 - 3*a^8 + 4*a^7 ˓→5*a^6 + 6*a^5 - 4*a^4 + 3*a^3 - 2*a^2 - 2*a + 4,
a^16 - a^15 - 3*a^14 - 4*a^13 - 4*a^12 - 3*a^11 - a^10 + 2*a^9 + 4*a^8 + 5*a^
˓→7 + 4*a^6 + 2*a^5 - 2*a^4 - 6*a^3 - 9*a^2 - 9*a - 7,
a^15 + a^14 + 2*a^11 + a^10 - a^9 + a^8 + 2*a^7 - a^5 + 2*a^3 - a^2 - 3*a +
˓→1,
5*a^16 - 6*a^14 + a^13 + 7*a^12 - 2*a^11 - 7*a^10 + 4*a^9 + 7*a^8 - 6*a^7 ˓→7*a^6 + 8*a^5 + 6*a^4 - 11*a^3 - 5*a^2 + 13*a + 4)
zeta ( n=2, all=False)
Return one, or a list of all, primitive n-th root of unity in this field.
INPUT:
•n - positive integer
•all - bool. If False (default), return a primitive 𝑛-th root of unity in this field, or raise a ValueError
exception if there are none. If True, return a list of all primitive 𝑛-th roots of unity in this field
(possibly empty).
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Note: To obtain the maximal order of a root of unity in this field, use self.number_of_roots_of_unity().
Note: We do not create the full unit group since that can be expensive, but we do use it if it is already
known.
EXAMPLES:
sage: K.<z> = NumberField(x^2 + 3)
sage: K.zeta(1)
1
sage: K.zeta(2)
-1
sage: K.zeta(2, all=True)
[-1]
sage: K.zeta(3)
1/2*z - 1/2
sage: K.zeta(3, all=True)
[1/2*z - 1/2, -1/2*z - 1/2]
sage: K.zeta(4)
Traceback (most recent call last):
...
ValueError: There are no 4th roots of unity in self.
sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2+1)
sage: K.zeta(4)
b
sage: K.zeta(4,all=True)
[b, -b]
sage: K.zeta(3)
Traceback (most recent call last):
...
ValueError: There are no 3rd roots of unity in self.
sage: K.zeta(3,all=True)
[]
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage: K.<a> = NumberField(1/2*x^2 + 1/6)
sage: K.zeta(3)
-3/2*a - 1/2
zeta_coefficients ( n)
Compute the first n coefficients of the Dedekind zeta function of this field as a Dirichlet series.
EXAMPLES:
sage: x = QQ['x'].0
sage: NumberField(x^2+1, 'a').zeta_coefficients(10)
[1, 1, 0, 1, 2, 0, 0, 1, 1, 2]
zeta_function ( prec=53, max_imaginary_part=0, max_asymp_coeffs=40)
Return the Zeta function of this number field.
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This actually returns an interface to Tim Dokchitser’s program for computing with the Dedekind zeta
function zeta_F(s) of the number field F.
INPUT:
•prec - integer (bits precision)
•max_imaginary_part - real number
•max_asymp_coeffs - integer
OUTPUT: The zeta function of this number field.
EXAMPLES:
sage: K.<a> = NumberField(ZZ['x'].0^2+ZZ['x'].0-1)
sage: Z = K.zeta_function()
sage: Z
Zeta function associated to Number Field in a with defining polynomial x^2 +
˓→x - 1
sage: Z(-1)
0.0333333333333333
sage: L.<a, b, c> = NumberField([x^2 - 5, x^2 + 3, x^2 + 1])
sage: Z = L.zeta_function()
sage: Z(5)
1.00199015670185
zeta_order ( )
Return the number of roots of unity in this field.
Note: We do not create the full unit group since that can be expensive, but we do use it if it is already
known.
EXAMPLES:
sage:
sage:
6
sage:
sage:
2
F.<alpha> = NumberField(x**22+3)
F.zeta_order()
F.<alpha> = NumberField(x**2-7)
F.zeta_order()
sage.rings.number_field.number_field. NumberField_generic_v1 ( poly, name, latex_name, canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field import NumberField_absolute_v1
sage: R.<x> = QQ[]
sage: NumberField_absolute_v1(x^2 + 1, 'i', 'i')
Number Field in i with defining polynomial x^2 + 1
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class sage.rings.number_field.number_field. NumberField_quadratic ( polynomial,
name=None,
latex_name=None,
check=True,
embedding=None,
assume_disc_small=False,
maximize_at_primes=None,
structure=None)
Bases: sage.rings.number_field.number_field.NumberField_absolute
Create a quadratic extension of the rational field.
√
The command QuadraticField(a) creates the field Q( 𝑎).
EXAMPLES:
sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3
sage: QuadraticField(-4, 'b')
Number Field in b with defining polynomial x^2 + 4
class_number ( proof=None)
Return the size of the class group of self.
If proof = False (not the default!) and the discriminant of the field is negative, then the following warning
from the PARI manual applies:
Warning: For 𝐷 < 0, this function may give incorrect results when the class group has a low
exponent (has many cyclic factors), because implementing Shank’s method in full generality slows it
down immensely.
EXAMPLES:
sage: QuadraticField(-23,'a').class_number()
3
√
These are all the primes so that the class number of Q( −𝑝) is 1:
sage: [d for d in prime_range(2,300) if not is_square(d) and QuadraticField(˓→d,'a').class_number() == 1]
[2, 3, 7, 11, 19, 43, 67, 163]
√
It is an open problem to prove that there are infinity many positive square-free 𝑑 such that Q( 𝑑) has class
number 1:
sage: len([d for d in range(2,200) if not is_square(d) and QuadraticField(d,'a
˓→').class_number() == 1])
121
discriminant ( v=None)
Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of
the trace pairing on the elements of the list v.
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INPUT:
•v (optional) - list of element of this number field
OUTPUT: Integer if v is omitted, and Rational otherwise.
EXAMPLES:
sage:
sage:
-4
sage:
sage:
-20
sage:
sage:
5
K.<i> = NumberField(x^2+1)
K.discriminant()
K.<a> = NumberField(x^2+5)
K.discriminant()
K.<a> = NumberField(x^2-5)
K.discriminant()
hilbert_class_field ( names)
Returns the Hilbert class field of this quadratic field as a relative extension of this field.
Note:
For the polynomial that defines this field as a relative extension, see the
hilbert_class_field_defining_polynomial command, which is vastly faster than
this command, since it doesn’t construct a relative extension.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 23)
sage: L = K.hilbert_class_field('b'); L
Number Field in b with defining polynomial x^3 - x^2 + 1 over its base field
sage: L.absolute_field('c')
Number Field in c with defining polynomial x^6 - 2*x^5 + 70*x^4 - 90*x^3 +
˓→1631*x^2 - 1196*x + 12743
sage: K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1
hilbert_class_field_defining_polynomial ( name=’x’)
Returns a polynomial over Q whose roots generate the Hilbert class field of this quadratic field as an
extension of this quadratic field.
Note: Computed using PARI via Schertz’s method. This implementation is quite fast.
EXAMPLES:
sage: K.<b> = QuadraticField(-23)
sage: K.hilbert_class_field_defining_polynomial()
x^3 - x^2 + 1
Note that this polynomial is
hilbert_class_polynomial :
not
the
actual
Hilbert
class
polynomial:
see
sage: K.hilbert_class_polynomial()
x^3 + 3491750*x^2 - 5151296875*x + 12771880859375
sage: K.<a> = QuadraticField(-431)
sage: K.class_number()
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21
sage: K.hilbert_class_field_defining_polynomial(name='z')
z^21 + 6*z^20 + 9*z^19 - 4*z^18 + 33*z^17 + 140*z^16 + 220*z^15 + 243*z^14 +
˓→297*z^13 + 461*z^12 + 658*z^11 + 743*z^10 + 722*z^9 + 681*z^8 + 619*z^7 +
˓→522*z^6 + 405*z^5 + 261*z^4 + 119*z^3 + 35*z^2 + 7*z + 1
hilbert_class_polynomial ( name=’x’)
Compute the Hilbert class polynomial of this quadratic field.
Right now, this is only implemented for imaginary quadratic fields.
EXAMPLES:
sage: K.<a> = QuadraticField(-3)
sage: K.hilbert_class_polynomial()
x
sage: K.<a> = QuadraticField(-31)
sage: K.hilbert_class_polynomial(name='z')
z^3 + 39491307*z^2 - 58682638134*z + 1566028350940383
is_galois ( )
Return True since all quadratic fields are automatically Galois.
EXAMPLES:
sage: QuadraticField(1234,'d').is_galois()
True
sage.rings.number_field.number_field. NumberField_quadratic_v1 ( poly,
name,
canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field import NumberField_quadratic_v1
sage: R.<x> = QQ[]
sage: NumberField_quadratic_v1(x^2 - 2, 'd')
Number Field in d with defining polynomial x^2 - 2
sage.rings.number_field.number_field. QuadraticField ( D, name=’a’, check=True,
embedding=True,
latex_name=’sqrt’, **args)
Return a quadratic field obtained by adjoining a square root of 𝐷 to the rational numbers, where 𝐷 is not a
perfect square.
INPUT:
•D - a rational number
•name - variable name (default: ‘a’)
•check - bool (default: True)
•embedding - bool or square root of D in an ambient field (default: True)
•latex_name - latex variable name (default: sqrt{D})
OUTPUT: A number field defined by a quadratic polynomial. Unless otherwise specified, it has an embedding
into R or C by sending the generator to the positive or upper-half-plane root.
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EXAMPLES:
sage: QuadraticField(3, 'a')
Number Field in a with defining polynomial x^2 - 3
sage: K.<theta> = QuadraticField(3); K
Number Field in theta with defining polynomial x^2 - 3
sage: RR(theta)
1.73205080756888
sage: QuadraticField(9, 'a')
Traceback (most recent call last):
...
ValueError: D must not be a perfect square.
sage: QuadraticField(9, 'a', check=False)
Number Field in a with defining polynomial x^2 - 9
Quadratic number fields derive from general number fields.
sage: from sage.rings.number_field.number_field import is_NumberField
sage: type(K)
<class 'sage.rings.number_field.number_field.NumberField_quadratic_with_category'>
sage: is_NumberField(K)
True
Quadratic number fields are cached:
sage: QuadraticField(-11, 'a') is QuadraticField(-11, 'a')
True
By default, quadratic fields come with a nice latex representation:
sage: K.<a> = QuadraticField(-7)
sage: latex(K)
\Bold{Q}(\sqrt{-7})
sage: latex(a)
\sqrt{-7}
sage: latex(1/(1+a))
-\frac{1}{8} \sqrt{-7} + \frac{1}{8}
sage: K.latex_variable_name()
'\\sqrt{-7}'
We can provide our own name as well:
sage: K.<a> = QuadraticField(next_prime(10^10), latex_name=r'\sqrt{D}')
sage: 1+a
a + 1
sage: latex(1+a)
\sqrt{D} + 1
sage: latex(QuadraticField(-1, 'a', latex_name=None).gen())
a
The name of the generator does not interfere with Sage preparser, see trac ticket #1135:
sage:
sage:
sage:
sage:
True
sage:
86
K1 = QuadraticField(5, 'x')
K2.<x> = QuadraticField(5)
K3.<x> = QuadraticField(5, 'x')
K1 is K2
K1 is K3
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True
sage: K1
Number Field in x with defining polynomial x^2 - 5
Note that, in presence of two different names for the generator, the name given by the preparser takes precedence:
sage: K4.<y> = QuadraticField(5, 'x'); K4
Number Field in y with defining polynomial x^2 - 5
sage: K1 == K4
False
sage.rings.number_field.number_field. is_AbsoluteNumberField ( x)
Return True if x is an absolute number field.
EXAMPLES:
sage: from sage.rings.number_field.number_field import is_AbsoluteNumberField
sage: is_AbsoluteNumberField(NumberField(x^2+1,'a'))
True
sage: is_AbsoluteNumberField(NumberField([x^3 + 17, x^2+1],'a'))
False
The rationals are a number field, but they’re not of the absolute number field class.
sage: is_AbsoluteNumberField(QQ)
False
sage.rings.number_field.number_field. is_CyclotomicField ( x)
Return True if x is a cyclotomic field, i.e., of the special cyclotomic field class. This function does not return
True for a number field that just happens to be isomorphic to a cyclotomic field.
EXAMPLES:
sage:
sage:
False
sage:
True
sage:
True
sage:
False
sage:
False
from sage.rings.number_field.number_field import is_CyclotomicField
is_CyclotomicField(NumberField(x^2 + 1,'zeta4'))
is_CyclotomicField(CyclotomicField(4))
is_CyclotomicField(CyclotomicField(1))
is_CyclotomicField(QQ)
is_CyclotomicField(7)
sage.rings.number_field.number_field. is_NumberFieldHomsetCodomain ( codomain)
Returns whether codomain
is a valid codomain for a number field homset.
This
is used by NumberField._Hom_ to determine whether the created homsets should be a
sage.rings.number_field.morphism.NumberFieldHomset .
EXAMPLES:
This currently accepts any parent (CC, RR, ...) in Fields :
sage: from sage.rings.number_field.number_field import is_
˓→NumberFieldHomsetCodomain
sage: is_NumberFieldHomsetCodomain(QQ)
True
sage: is_NumberFieldHomsetCodomain(NumberField(x^2 + 1, 'x'))
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True
sage:
False
sage:
False
sage:
False
sage:
False
is_NumberFieldHomsetCodomain(ZZ)
is_NumberFieldHomsetCodomain(3)
is_NumberFieldHomsetCodomain(MatrixSpace(QQ, 2))
is_NumberFieldHomsetCodomain(InfinityRing)
Question: should, for example, QQ-algebras be accepted as well?
Caveat: Gap objects are not (yet) in Fields , and therefore not accepted as number field homset codomains:
sage: is_NumberFieldHomsetCodomain(gap.Rationals)
False
sage.rings.number_field.number_field. is_QuadraticField ( x)
Return True if x is of the quadratic number field type.
EXAMPLES:
sage:
sage:
True
sage:
True
sage:
False
from sage.rings.number_field.number_field import is_QuadraticField
is_QuadraticField(QuadraticField(5,'a'))
is_QuadraticField(NumberField(x^2 - 5, 'b'))
is_QuadraticField(NumberField(x^3 - 5, 'b'))
A quadratic field specially refers to a number field, not a finite field:
sage: is_QuadraticField(GF(9,'a'))
False
sage.rings.number_field.number_field. is_fundamental_discriminant ( D)
Return True if the integer 𝐷 is a fundamental discriminant, i.e., if 𝐷 ∼
= 0, 1 (mod 4), and 𝐷 ̸= 0, 1 and either
(1) 𝐷 is square free or (2) we have 𝐷 ∼
= 0 (mod 4) with 𝐷/4 ∼
= 2, 3 (mod 4) and 𝐷/4 square free. These are
exactly the discriminants of quadratic fields.
EXAMPLES:
sage: [D for D
[-15, -11, -8,
sage: [D for D
˓→disc() == D]
[-15, -11, -8,
in range(-15,15) if is_fundamental_discriminant(D)]
-7, -4, -3, 5, 8, 12, 13]
in range(-15,15) if not is_square(D) and QuadraticField(D,'a').
-7, -4, -3, 5, 8, 12, 13]
sage.rings.number_field.number_field. proof_flag ( t)
Used for easily determining the correct proof flag to use.
Returns t if t is not None, otherwise returns the system-wide proof-flag for number fields (default: True).
EXAMPLES:
sage: from sage.rings.number_field.number_field import proof_flag
sage: proof_flag(True)
True
sage: proof_flag(False)
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False
sage: proof_flag(None)
True
sage: proof_flag("banana")
'banana'
sage.rings.number_field.number_field. put_natural_embedding_first ( v)
Helper function for embeddings() functions for number fields.
INPUT: a list of embeddings of a number field
OUTPUT: None. The list is altered in-place, so that, if possible, the first embedding has been switched with one
of the others, so that if there is an embedding which preserves the generator names then it appears first.
EXAMPLES:
sage: K.<a> = CyclotomicField(7)
sage: embs = K.embeddings(K)
sage: [e(a) for e in embs] # random - there is no natural sort order
[a, a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1]
sage: id = [ e for e in embs if e(a) == a ][0]; id
Ring endomorphism of Cyclotomic Field of order 7 and degree 6
Defn: a |--> a
sage: permuted_embs = list(embs); permuted_embs.remove(id); permuted_embs.
˓→append(id)
sage: [e(a) for e in permuted_embs] # random - but natural map is not first
[a^2, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a]
sage: permuted_embs[0] != a
True
sage: from sage.rings.number_field.number_field import put_natural_embedding_first
sage: put_natural_embedding_first(permuted_embs)
sage: [e(a) for e in permuted_embs] # random - but natural map is first
[a, a^3, a^4, a^5, -a^5 - a^4 - a^3 - a^2 - a - 1, a^2]
sage: permuted_embs[0] == id
True
sage.rings.number_field.number_field. refine_embedding ( e, prec=None)
Given an embedding from a number field to either R or C, returns an equivalent embedding with higher precision.
INPUT:
•e - an embedding of a number field into either RR or CC (with some precision)
•prec - (default None) the desired precision; if None, current precision is doubled; if Infinity, the
equivalent embedding into either QQbar or AA is returned.
EXAMPLES:
sage:
sage:
sage:
sage:
10
sage:
sage:
25
from sage.rings.number_field.number_field import refine_embedding
K = CyclotomicField(3)
e10 = K.complex_embedding(10)
e10.codomain().precision()
e25 = refine_embedding(e10, prec=25)
e25.codomain().precision()
An example where we extend a real embedding into AA :
1.1. Number Fields
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sage: K.<a> = NumberField(x^3-2)
sage: K.signature()
(1, 1)
sage: e = K.embeddings(RR)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Real Field with 53 bits of precision
Defn: a |--> 1.25992104989487
sage: e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Algebraic Real Field
Defn: a |--> 1.259921049894873?
Now we can obtain arbitrary precision values with no trouble:
sage: RealField(150)(e(a))
1.2599210498948731647672106072782283505702515
sage: _^3
2.0000000000000000000000000000000000000000000
sage: RealField(200)(e(a^2-3*a+7))
4.8076379022835799804500738174376232086807389337953290695624
Complex embeddings can be extended into QQbar :
sage: e = K.embeddings(CC)[0]; e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Complex Field with 53 bits of precision
Defn: a |--> -0.62996052494743... - 1.09112363597172*I
sage: e = refine_embedding(e,Infinity); e
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 2
To:
Algebraic Field
Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I
sage: ComplexField(200)(e(a))
-0.62996052494743658238360530363911417528512573235075399004099 - 1.
˓→0911236359717214035600726141898088813258733387403009407036*I
sage: e(a)^3
2
Embeddings into lazy fields work:
sage: L = CyclotomicField(7)
sage: x = L.specified_complex_embedding(); x
Generic morphism:
From: Cyclotomic Field of order 7 and degree 6
To:
Complex Lazy Field
Defn: zeta7 -> 0.623489801858734? + 0.781831482468030?*I
sage: refine_embedding(x, 300)
Ring morphism:
From: Cyclotomic Field of order 7 and degree 6
To:
Complex Field with 300 bits of precision
Defn: zeta7 |--> 0.
˓→623489801858733530525004884004239810632274730896402105365549439096853652456487284575942507
˓→+ 0.
˓→781831482468029808708444526674057750232334518708687528980634958045091731633936441700868007*I
sage: refine_embedding(x, infinity)
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Ring morphism:
From: Cyclotomic Field of order 7 and degree 6
To:
Algebraic Field
Defn: zeta7 |--> 0.6234898018587335? + 0.7818314824680299?*I
When the old embedding is into the real lazy field, then only real embeddings should be considered. See trac
ticket #17495:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3 + x - 1, embedding=0.68)
sage: from sage.rings.number_field.number_field import refine_embedding
sage: refine_embedding(K.specified_complex_embedding(), 100)
Ring morphism:
From: Number Field in a with defining polynomial x^3 + x - 1
To:
Real Field with 100 bits of precision
Defn: a |--> 0.68232780382801932736948373971
sage: refine_embedding(K.specified_complex_embedding(), Infinity)
Ring morphism:
From: Number Field in a with defining polynomial x^3 + x - 1
To:
Algebraic Real Field
Defn: a |--> 0.6823278038280193?
1.2 Base class for all number fields
class sage.rings.number_field.number_field_base. NumberField
Bases: sage.rings.ring.Field
Base class for all number fields.
OK ( *args, **kwds)
Synomym for self.maximal_order(...) .
EXAMPLES:
sage: NumberField(x^3 - 2,'a').OK()
Maximal Order in Number Field in a with defining polynomial x^3 - 2
bach_bound ( )
Return the Bach bound associated to this number field. Assuming the General Riemann Hypothesis, this
is a bound B so that every integral ideal is equivalent modulo principal fractional ideals to an integral ideal
of norm at most B.
See also:
minkowski_bound()
OUTPUT:
symbolic expression or the Integer 1
EXAMPLES:
We compute both the Minkowski and Bach bounds for a quadratic field, where the Minkowski bound is
much better:
sage: K = QQ[sqrt(5)]
sage: K.minkowski_bound()
1/2*sqrt(5)
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sage: K.minkowski_bound().n()
1.11803398874989
sage: K.bach_bound()
12*log(5)^2
sage: K.bach_bound().n()
31.0834847277628
We compute both the Minkowski and Bach bounds for a bigger degree field, where the Bach bound is
much better:
sage: K = CyclotomicField(37)
sage: K.minkowski_bound().n()
7.50857335698544e14
sage: K.bach_bound().n()
191669.304126267
The bound of course also works for the rational numbers: sage: QQ.minkowski_bound() 1
degree ( )
Return the degree of this number field.
EXAMPLES:
sage: NumberField(x^3 + 9, 'a').degree()
3
discriminant ( )
Return the discriminant of this number field.
EXAMPLES:
sage: NumberField(x^3 + 9, 'a').discriminant()
-243
is_absolute ( )
Return True if self is viewed as a single extension over Q.
EXAMPLES:
sage:
sage:
True
sage:
sage:
sage:
False
sage:
True
K.<a> = NumberField(x^3+2)
K.is_absolute()
y = polygen(K)
L.<b> = NumberField(y^2+1)
L.is_absolute()
QQ.is_absolute()
is_finite ( )
Return False since number fields are not finite.
EXAMPLES:
sage: z = polygen(QQ)
sage: K.<theta, beta> = NumberField([z^3 - 3, z^2 + 1])
sage: K.is_finite()
False
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sage: K.order()
+Infinity
maximal_order ( )
Return the maximal order, i.e., the ring of integers of this number field.
EXAMPLES:
sage: NumberField(x^3 - 2,'b').maximal_order()
Maximal Order in Number Field in b with defining polynomial x^3 - 2
minkowski_bound ( )
Return the Minkowski bound associated to this number field, which is a bound B so that every integral
ideal is equivalent modulo principal fractional ideals to an integral ideal of norm at most B.
See also:
bach_bound()
OUTPUT:
symbolic expression or Rational
EXAMPLES:
The Minkowski bound for Q[𝑖] tells us that the class number is 1:
sage: K = QQ[I]
sage: B = K.minkowski_bound(); B
4/pi
sage: B.n()
1.27323954473516
√
We compute the Minkowski bound for Q[ 3 2]:
sage: K = QQ[2^(1/3)]
sage: B = K.minkowski_bound(); B
16/3*sqrt(3)/pi
sage: B.n()
2.94042077558289
sage: int(B)
2
√
We compute the Minkowski bound for Q[ 10], which has class number 2:
sage: K = QQ[sqrt(10)]
sage: B = K.minkowski_bound(); B
sqrt(10)
sage: int(B)
3
sage: K.class_number()
2
√
√
We compute the Minkowski bound for Q[ 2 + 3]:
sage:
sage:
sage:
9/2
sage:
K.<y,z> = NumberField([x^2-2, x^2-3])
L.<w> = QQ[sqrt(2) + sqrt(3)]
B = K.minkowski_bound(); B
int(B)
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4
sage: B == L.minkowski_bound()
True
sage: K.class_number()
1
The bound of course also works for the rational numbers:
sage: QQ.minkowski_bound()
1
ring_of_integers ( *args, **kwds)
Synomym for self.maximal_order(...) .
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 1)
sage: K.ring_of_integers()
Gaussian Integers in Number Field in a with defining polynomial x^2 + 1
signature ( )
Return (r1, r2), where r1 and r2 are the number of real embeddings and pairs of complex embeddings of
this field, respectively.
EXAMPLES:
sage: NumberField(x^3 - 2, 'a').signature()
(1, 1)
sage.rings.number_field.number_field_base. is_NumberField ( x)
Return True if x is of number field type.
EXAMPLES:
sage:
sage:
True
sage:
True
sage:
True
from sage.rings.number_field.number_field_base import is_NumberField
is_NumberField(NumberField(x^2+1,'a'))
is_NumberField(QuadraticField(-97,'theta'))
is_NumberField(CyclotomicField(97))
Note that the rational numbers QQ are a number field.:
sage: is_NumberField(QQ)
True
sage: is_NumberField(ZZ)
False
1.3 Relative Number Fields
AUTHORS:
• William Stein (2004, 2005): initial version
• Steven Sivek (2006-05-12): added support for relative extensions
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• William Stein (2007-09-04): major rewrite and documentation
• Robert Bradshaw (2008-10): specified embeddings into ambient fields
• Nick Alexander (2009-01): modernize coercion implementation
• Robert Harron (2012-08): added is_CM_extension
• Julian Rueth (2014-04-03): absolute number fields are unique parents
This example follows one in the Magma reference manual:
sage: K.<y> = NumberField(x^4 - 420*x^2 + 40000)
sage: z = y^5/11; z
420/11*y^3 - 40000/11*y
sage: R.<y> = PolynomialRing(K)
sage: f = y^2 + y + 1
sage: L.<a> = K.extension(f); L
Number Field in a with defining polynomial y^2 + y + 1 over its base field
sage: KL.<b> = NumberField([x^4 - 420*x^2 + 40000, x^2 + x + 1]); KL
Number Field in b0 with defining polynomial x^4 - 420*x^2 + 40000 over its base field
We do some arithmetic in a tower of relative number fields:
sage: K.<cuberoot2> = NumberField(x^3 - 2)
sage: L.<cuberoot3> = K.extension(x^3 - 3)
sage: S.<sqrt2> = L.extension(x^2 - 2)
sage: S
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
sage: sqrt2 * cuberoot3
cuberoot3*sqrt2
sage: (sqrt2 + cuberoot3)^5
(20*cuberoot3^2 + 15*cuberoot3 + 4)*sqrt2 + 3*cuberoot3^2 + 20*cuberoot3 + 60
sage: cuberoot2 + cuberoot3
cuberoot3 + cuberoot2
sage: cuberoot2 + cuberoot3 + sqrt2
sqrt2 + cuberoot3 + cuberoot2
sage: (cuberoot2 + cuberoot3 + sqrt2)^2
(2*cuberoot3 + 2*cuberoot2)*sqrt2 + cuberoot3^2 + 2*cuberoot2*cuberoot3 + cuberoot2^2
˓→+ 2
sage: cuberoot2 + sqrt2
sqrt2 + cuberoot2
sage: a = S(cuberoot2); a
cuberoot2
sage: a.parent()
Number Field in sqrt2 with defining polynomial x^2 - 2 over its base field
WARNING: Doing arithmetic in towers of relative fields that depends on canonical coercions is currently VERY
SLOW. It is much better to explicitly coerce all elements into a common field, then do arithmetic with them there
(which is quite fast).
sage.rings.number_field.number_field_rel. NumberField_extension_v1 ( base_field,
poly,
name, latex_name,
canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
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sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1
sage: R.<x> = CyclotomicField(3)[]
sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
Number Field in a with defining polynomial x^2 + 7 over its base field
class sage.rings.number_field.number_field_rel. NumberField_relative ( base,
polynomial,
name, latex_name=None,
names=None,
check=True,
embedding=None,
structure=None)
Bases: sage.rings.number_field.number_field.NumberField_generic
INPUT:
•base – the base field
•polynomial – a polynomial which must be defined in the ring 𝐾[𝑥], where 𝐾 is the base field.
•name – a string, the variable name
•latex_name – a string or None (default: None ), variable name for latex printing
•check – a boolean (default: True ), whether to check irreducibility of polynomial
•embedding – currently not supported, must be None
•structure – an instance of structure.NumberFieldStructure or None (default: None ),
provides additional information about this number field, e.g., the absolute number field from which it was
created
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2)
sage: t = polygen(K)
sage: L.<b> = K.extension(t^2+t+a); L
Number Field in b with defining polynomial x^2 + x + a over its base field
absolute_base_field ( )
Return the base field of this relative extension, but viewed as an absolute field over Q.
EXAMPLES:
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 3, x^3 + 2])
sage: K
Number Field in a with defining polynomial x^2 + 2 over its base field
sage: K.base_field()
Number Field in b with defining polynomial x^3 + 3 over its base field
sage: K.absolute_base_field()[0]
Number Field in a0 with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
sage: K.base_field().absolute_field('z')
Number Field in z with defining polynomial x^9 + 3*x^6 + 165*x^3 + 1
absolute_degree ( )
The degree of this relative number field over the rational field.
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EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
sage: K.absolute_degree()
6
absolute_different ( )
Return the absolute different of this relative number field 𝐿, as an ideal of 𝐿. To get the relative different
of 𝐿/𝐾, use L.relative_different() .
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: t = K['t'].gen()
sage: L.<b> = K.extension(t^4 - i)
sage: L.absolute_different()
Fractional ideal (8)
absolute_discriminant ( v=None)
Return the absolute discriminant of this relative number field or if v is specified, the determinant of the
trace pairing on the elements of the list v .
INPUT:
•v (optional) – list of element of this relative number field.
OUTPUT: Integer if v is omitted, and Rational otherwise.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: t = K['t'].gen()
sage: L.<b> = K.extension(t^4 - i)
sage: L.absolute_discriminant()
16777216
sage: L.absolute_discriminant([(b + i)^j for j in range(8)])
61911970349056
absolute_field ( names)
Return an absolute number field 𝐾 that is isomorphic to this field along with a field-theoretic bijection
from self to 𝐾 and from 𝐾 to self.
INPUT:
•names – string; name of generator of the absolute field
OUTPUT: an absolute number field
Also, K.structure() returns from_K and to_K , where from_K is an isomorphism from 𝐾 to
self and to_K is an isomorphism from self to 𝐾.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]);
Number Field in a with defining polynomial x^4 +
sage: L.<xyz> = K.absolute_field(); L
Number Field in xyz with defining polynomial x^8
˓→49
sage: L.<c> = K.absolute_field(); L
Number Field in c with defining polynomial x^8 +
1.3. Relative Number Fields
K
3 over its base field
+ 8*x^6 + 30*x^4 - 40*x^2 +
8*x^6 + 30*x^4 - 40*x^2 + 49
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sage: from_L, to_L = L.structure()
sage: from_L
Isomorphism map:
From: Number Field in c with defining
˓→40*x^2 + 49
To:
Number Field in a with defining
sage: from_L(c)
a - b
sage: to_L
Isomorphism map:
From: Number Field in a with defining
To:
Number Field in c with defining
˓→40*x^2 + 49
sage: to_L(a)
-5/182*c^7 - 87/364*c^5 - 185/182*c^3 +
sage: to_L(b)
-5/182*c^7 - 87/364*c^5 - 185/182*c^3 sage: to_L(a)^4
-3
sage: to_L(b)^2
-2
polynomial x^8 + 8*x^6 + 30*x^4 polynomial x^4 + 3 over its base field
polynomial x^4 + 3 over its base field
polynomial x^8 + 8*x^6 + 30*x^4 -
323/364*c
41/364*c
absolute_generator ( )
Return the chosen generator over Q for this relative number field.
EXAMPLES:
sage: y = polygen(QQ,'y')
sage: k.<a> = NumberField([y^2 + 2, y^4 + 3])
sage: g = k.absolute_generator(); g
a0 - a1
sage: g.minpoly()
x^2 + 2*a1*x + a1^2 + 2
sage: g.absolute_minpoly()
x^8 + 8*x^6 + 30*x^4 - 40*x^2 + 49
absolute_polynomial ( )
Return the polynomial over Q that defines this field as an extension of the rational numbers.
Note: The absolute polynomial of a relative number field is chosen to be equal to the defining polynomial
of the underlying PARI absolute number field (it cannot be specified by the user). In particular, it is always
a monic polynomial with integral coefficients. On the other hand, the defining polynomial of an absolute
number field and the relative polynomial of a relative number field are in general different from their PARI
counterparts.
EXAMPLES:
sage: k.<a, b> = NumberField([x^2 + 1, x^3 + x + 1]); k
Number Field in a with defining polynomial x^2 + 1 over its base field
sage: k.absolute_polynomial()
x^6 + 5*x^4 - 2*x^3 + 4*x^2 + 4*x + 1
An example comparing the various defining polynomials to their PARI counterparts:
sage: k.<a, c> = NumberField([x^2 + 1/3, x^2 + 1/4])
sage: k.absolute_polynomial()
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x^4 sage:
x^4 sage:
x^2 +
sage:
y^2 +
sage:
x^2 +
sage:
x^2 +
x^2 + 1
k.pari_polynomial()
x^2 + 1
k.base_field().absolute_polynomial()
1/4
k.pari_absolute_base_polynomial()
1
k.relative_polynomial()
1/3
k.pari_relative_polynomial()
Mod(y, y^2 + 1)*x - 1
absolute_polynomial_ntl ( )
Return defining polynomial of this number field as a pair, an ntl polynomial and a denominator.
This is used mainly to implement some internal arithmetic.
EXAMPLES:
sage: NumberField(x^2 + (2/3)*x - 9/17,'a').absolute_polynomial_ntl()
([-27 34 51], 51)
absolute_vector_space ( )
Return vector space over Q of self and isomorphisms from the vector space to self and in the other direction.
EXAMPLES:
sage: K.<a,b> = NumberField([x^3 + 3, x^3 + 2]); K
Number Field in a with defining polynomial x^3 + 3 over its base field
sage: V,from_V,to_V = K.absolute_vector_space(); V
Vector space of dimension 9 over Rational Field
sage: from_V
Isomorphism map:
From: Vector space of dimension 9 over Rational Field
To:
Number Field in a with defining polynomial x^3 + 3 over its base field
sage: to_V
Isomorphism map:
From: Number Field in a with defining polynomial x^3 + 3 over its base field
To:
Vector space of dimension 9 over Rational Field
sage: c = (a+1)^5; c
7*a^2 - 10*a - 29
sage: to_V(c)
(-29, -712/9, 19712/45, 0, -14/9, 364/45, 0, -4/9, 119/45)
sage: from_V(to_V(c))
7*a^2 - 10*a - 29
sage: from_V(3*to_V(b))
3*b
automorphisms ( )
Compute all Galois automorphisms of self over the base field. This is different than computing the embeddings of self into self; there, automorphisms that do not fix the base field are considered.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 10000, x^2 + x + 50]); K
Number Field in a with defining polynomial x^2 + 10000 over its base field
sage: K.automorphisms()
[
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Relative number field endomorphism of Number Field in a with defining
˓→polynomial x^2 + 10000 over its base field
Defn: a |--> a
b |--> b,
Relative number field endomorphism of Number Field in a with defining
˓→polynomial x^2 + 10000 over its base field
Defn: a |--> -a
b |--> b
]
sage: rho, tau = K.automorphisms()
sage: tau(a)
-a
sage: tau(b) == b
True
sage: L.<b, a> = NumberField([x^2 + x + 50, x^2 + 10000, ]);
Number Field in b with defining polynomial x^2 + x + 50 over
sage: L.automorphisms()
[
Relative number field endomorphism of Number Field in b with
˓→polynomial x^2 + x + 50 over its base field
Defn: b |--> b
a |--> a,
Relative number field endomorphism of Number Field in b with
˓→polynomial x^2 + x + 50 over its base field
Defn: b |--> -b - 1
a |--> a
]
sage: rho, tau = L.automorphisms()
sage: tau(a) == a
True
sage: tau(b)
-b - 1
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.automorphisms()
[
Relative number field endomorphism of Number Field in c
˓→polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base
Defn: c |--> c
a |--> a
b |--> b,
Relative number field endomorphism of Number Field in c
˓→polynomial Y^2 + (-2*b - 3)*a - 2*b - 6 over its base
Defn: c |--> -c
a |--> a
b |--> b
]
L
its base field
defining
defining
with defining
field
with defining
field
base_field ( )
Return the base field of this relative number field.
EXAMPLES:
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sage: k.<a> = NumberField([x^3 + x + 1])
sage: R.<z> = k[]
sage: L.<b> = NumberField(z^3 + a)
sage: L.base_field()
Number Field in a with defining polynomial x^3 + x + 1
sage: L.base_field() is k
True
This is very useful because the print representation of a relative field doesn’t describe the base field.:
sage: L
Number Field in b with defining polynomial z^3 + a over its base field
base_ring ( )
This is exactly the same as base_field.
EXAMPLES:
sage: k.<a> = NumberField([x^2 + 1, x^3 + x + 1])
sage: k.base_ring()
Number Field in a1 with defining polynomial x^3 + x + 1
sage: k.base_field()
Number Field in a1 with defining polynomial x^3 + x + 1
change_names ( names)
Return relative number field isomorphic to self but with the given generator names.
INPUT:
•names – number of names should be at most the number of generators of self, i.e., the number of
steps in the tower of relative fields.
Also, K.structure() returns from_K and to_K , where from_K is an isomorphism from 𝐾 to self
and to_K is an isomorphism from self to 𝐾.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]);
Number Field in a with defining polynomial x^4 +
sage: L.<c,d> = K.change_names()
sage: L
Number Field in c with defining polynomial x^4 +
sage: L.base_field()
Number Field in d with defining polynomial x^2 +
K
3 over its base field
3 over its base field
2
An example with a 3-level tower:
sage: K.<a,b,c> = NumberField([x^2 + 17, x^2 +
Number Field in a with defining polynomial x^2
sage: L.<m,n,r> = K.change_names()
sage: L
Number Field in m with defining polynomial x^2
sage: L.base_field()
Number Field in n with defining polynomial x^2
sage: L.base_field().base_field()
Number Field in r with defining polynomial x^3
x + 1, x^3 - 2]); K
+ 17 over its base field
+ 17 over its base field
+ x + 1 over its base field
- 2
And a more complicated example:
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sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: L.<m, n, r> = K.change_names(); L
Number Field in m with defining polynomial x^2 + (-2*r
˓→its base field
sage: L.structure()
(Isomorphism given by variable name change map:
From: Number Field in m with defining polynomial x^2
˓→6 over its base field
To:
Number Field in c with defining polynomial Y^2
˓→6 over its base field,
Isomorphism given by variable name change map:
From: Number Field in c with defining polynomial Y^2
˓→6 over its base field
To:
Number Field in m with defining polynomial x^2
˓→6 over its base field)
- 3)*n - 2*r - 6 over
+ (-2*r - 3)*n - 2*r + (-2*b - 3)*a - 2*b -
+ (-2*b - 3)*a - 2*b + (-2*r - 3)*n - 2*r -
composite_fields ( other, names=None, both_maps=False, preserve_embedding=True)
List of all possible composite number fields formed from self and other, together with (optionally) embeddings into the compositum; see the documentation for both_maps below.
Since relative fields do not have ambient embeddings, preserve_embedding has no effect. In every case all
possible composite number fields are returned.
INPUT:
•other - a number field
•names - generator name for composite fields
•both_maps - (default: False) if True, return quadruples (F, self_into_F, other_into_F, k) such that
self_into_F maps self into F, other_into_F maps other into F. For relative number fields k is always
None.
•preserve_embedding - (default: True) has no effect, but is kept for compatibility with the
absolute version of this function. In every case the list of all possible compositums is returned.
OUTPUT:
•list - list of the composite fields, possibly with maps.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 5, x^2 - 2])
sage: L.<c, d> = NumberField([x^2 + 5, x^2 - 3])
sage: K.composite_fields(L, 'e')
[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^2
˓→+ 25600]
sage: K.composite_fields(L, 'e', both_maps=True)
[[Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 + 3840*x^
˓→2 + 25600,
Relative number field morphism:
From: Number Field in a with defining polynomial x^2 + 5 over its base field
To:
Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 +
˓→3840*x^2 + 25600
Defn: a |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
b |--> -21/166400*e^7 + 73/20800*e^5 - 779/10400*e^3 + 7/260*e,
Relative number field morphism:
From: Number Field in c with defining polynomial x^2 + 5 over its base field
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To:
Number Field in e with defining polynomial x^8 - 24*x^6 + 464*x^4 +
3840*x^2 + 25600
Defn: c |--> -9/66560*e^7 + 11/4160*e^5 - 241/4160*e^3 - 101/104*e
d |--> -3/25600*e^7 + 7/1600*e^5 - 147/1600*e^3 + 1/40*e,
None]]
˓→
defining_polynomial ( )
Return the defining polynomial of this relative number field.
This is exactly the same as relative_polynomial() .
EXAMPLES:
sage: C.<z> = CyclotomicField(5)
sage: PC.<X> = C[]
sage: K.<a> = C.extension(X^2 + X + z); K
Number Field in a with defining polynomial X^2 + X + z over its base field
sage: K.defining_polynomial()
X^2 + X + z
degree ( )
The degree, unqualified, of a relative number field is deliberately not implemented, so that a user cannot
mistake the absolute degree for the relative degree, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
sage: K.degree()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field you must use relative_degree
˓→or absolute_degree as appropriate
different ( )
The different, unqualified, of a relative number field is deliberately not implemented, so that a user cannot
mistake the absolute different for the relative different, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
sage: K.different()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field you must use relative_
˓→different or absolute_different as appropriate
disc ( )
The discriminant, unqualified, of a relative number field is deliberately not implemented, so that a user
cannot mistake the absolute discriminant for the relative discriminant, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
sage: K.disc()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field you must use relative_
˓→discriminant or absolute_discriminant as appropriate
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discriminant ( )
The discriminant, unqualified, of a relative number field is deliberately not implemented, so that a user
cannot mistake the absolute discriminant for the relative discriminant, or vice versa.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
sage: K.discriminant()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field you must use relative_
˓→discriminant or absolute_discriminant as appropriate
embeddings ( K)
Compute all field embeddings of the relative number field self into the field 𝐾 (which need not even be a
number field, e.g., it could be the complex numbers). This will return an identical result when given 𝐾 as
input again.
If possible, the most natural embedding of self into 𝐾 is put first in the list.
INPUT:
•K – a field
EXAMPLES:
sage: K.<a,b> = NumberField([x^3 - 2, x^2+1])
sage: f = K.embeddings(ComplexField(58)); f
[
Relative number field morphism:
From: Number Field in a with defining polynomial x^3 - 2 over its base field
To:
Complex Field with 58 bits of precision
Defn: a |--> -0.62996052494743676 - 1.0911236359717214*I
b |--> -1.9428902930940239e-16 + 1.0000000000000000*I,
...
Relative number field morphism:
From: Number Field in a with defining polynomial x^3 - 2 over its base field
To:
Complex Field with 58 bits of precision
Defn: a |--> 1.2599210498948731
b |--> -0.99999999999999999*I
]
sage: f[0](a)^3
2.0000000000000002 - 8.6389229103644993e-16*I
sage: f[0](b)^2
-1.0000000000000001 - 3.8857805861880480e-16*I
sage: f[0](a+b)
-0.62996052494743693 - 0.091123635971721295*I
galois_closure ( names=None)
Return the absolute number field 𝐾 that is the Galois closure of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
Number Field in a with defining polynomial x^4 + 3 over its base field
sage: K.galois_closure('c')
Number Field in c with defining polynomial x^16 + 16*x^14 + 28*x^12 + 784*x^
˓→10 + 19846*x^8 - 595280*x^6 + 2744476*x^4 + 3212848*x^2 + 29953729
galois_group ( type=’pari’, algorithm=’pari’, names=None)
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Return the Galois group of the Galois closure of this number field as an abstract group. Note that even
though this is an extension 𝐿/𝐾, the group will be computed as if it were 𝐿/Q.
INPUT:
•type - 'pari' or 'gap' : type of object to return – a wrapper around a Pari or Gap transitive
group object. •algorithm - ‘pari’, ‘kash’, ‘magma’ (default: ‘pari’, except when the degree is >= 12 when ‘kash’ is
tried)
At present much less functionality is available for Galois groups of relative extensions than absolute ones,
so try the galois_group method of the corresponding absolute field.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2 + 1)
sage: R.<t> = PolynomialRing(K)
sage: L = K.extension(t^5-t+a, 'b')
sage: L.galois_group(type="pari")
Galois group PARI group [240, -1, 22, "S(5)[x]2"] of degree 10 of the Number
˓→Field in b with defining polynomial t^5 - t + a over its base field
gen ( n=0)
Return the 𝑛‘th generator of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
Number Field in a with defining polynomial x^4 + 3 over its base field
sage: K.gens()
(a, b)
sage: K.gen(0)
a
gens ( )
Return the generators of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
Number Field in a with defining polynomial x^4 + 3 over its base field
sage: K.gens()
(a, b)
is_CM_extension ( )
Return True is this is a CM extension, i.e. a totally imaginary quadratic extension of a totally real field.
EXAMPLES:
sage:
sage:
sage:
True
sage:
sage:
sage:
True
sage:
sage:
F.<a> = NumberField(x^2 - 5)
K.<z> = F.extension(x^2 + 7)
K.is_CM_extension()
K = CyclotomicField(7)
K_rel = K.relativize(K.gen() + K.gen()^(-1), 'z')
K_rel.is_CM_extension()
F = CyclotomicField(3)
K.<z> = F.extension(x^3 - 2)
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sage: K.is_CM_extension()
False
A CM field K such that K/F is not a CM extension
sage:
sage:
sage:
False
sage:
True
F.<a> = NumberField(x^2 + 1)
K.<z> = F.extension(x^2 - 3)
K.is_CM_extension()
K.is_CM()
is_absolute ( )
Returns False, since this is not an absolute field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
Number Field in a with defining polynomial x^4 + 3 over its base field
sage: K.is_absolute()
False
sage: K.is_relative()
True
is_free ( proof=None)
Determine whether or not 𝐿/𝐾 is free (i.e. if 𝒪𝐿 is a free 𝒪𝐾 -module).
INPUT:
•proof – default: True
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
False
x = polygen(QQ)
K.<a> = NumberField(x^2+6)
x = polygen(K)
L.<b> = K.extension(x^2 + 3)
L.is_free()
## extend by x^2+3
is_galois ( )
For a relative number field, is_galois() is deliberately not implemented, since it is not clear
whether this would mean “Galois over Q” or “Galois over the given base field”. Use either
is_galois_absolute() or is_galois_relative() respectively.
EXAMPLES:
sage: k.<a> =NumberField([x^3 - 2, x^2 + x + 1])
sage: k.is_galois()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field L you must use either L.is_
˓→galois_relative() or L.is_galois_absolute() as appropriate
is_galois_absolute ( )
Return True if for this relative extension 𝐿/𝐾, 𝐿 is a Galois extension of Q.
EXAMPLES:
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sage: K.<a> = NumberField(x^3 - 2)
sage: y = polygen(K); L.<b> = K.extension(y^2 - a)
sage: L.is_galois_absolute()
False
is_galois_relative ( )
Return True if for this relative extension 𝐿/𝐾, 𝐿 is a Galois extension of 𝐾.
EXAMPLES:
sage:
sage:
sage:
sage:
True
sage:
sage:
False
K.<a> = NumberField(x^3 - 2)
y = polygen(K)
L.<b> = K.extension(y^2 - a)
L.is_galois_relative()
M.<c> = K.extension(y^3 - a)
M.is_galois_relative()
The next example previously gave a wrong result; see trac ticket #9390:
sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: F.is_galois_relative()
True
is_isomorphic_relative ( other, base_isom=None)
For this relative extension 𝐿/𝐾 and another relative extension 𝑀/𝐾, return True if there is a 𝐾-linear
isomorphism from 𝐿 to 𝑀 . More generally, other can be a relative extension 𝑀/𝐾 ′ with base_isom
an isomorphism from 𝐾 to 𝐾 ′ .
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
False
sage:
True
sage:
sage:
False
K.<z9> = NumberField(x^6 + x^3 + 1)
R.<z> = PolynomialRing(K)
m1 = 3*z9^4 - 4*z9^3 - 4*z9^2 + 3*z9 - 8
L1 = K.extension(z^2 - m1, 'b1')
G = K.galois_group(); gamma = G.gen()
m2 = (gamma^2)(m1)
L2 = K.extension(z^2 - m2, 'b2')
L1.is_isomorphic_relative(L2)
L1.is_isomorphic(L2)
L3 = K.extension(z^4 - m1, 'b3')
L1.is_isomorphic_relative(L3)
If we have two extensions over different, but isomorphic, bases, we can compare them by letting
base_isom be an isomorphism from self’s base field to other’s base field:
sage:
sage:
sage:
sage:
sage:
sage:
True
sage:
Kcyc.<zeta9> = CyclotomicField(9)
Rcyc.<zcyc> = PolynomialRing(Kcyc)
phi1 = K.hom([zeta9])
m1cyc = phi1(m1)
L1cyc = Kcyc.extension(zcyc^2 - m1cyc, 'b1cyc')
L1.is_isomorphic_relative(L1cyc, base_isom=phi1)
L2.is_isomorphic_relative(L1cyc, base_isom=phi1)
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False
sage: phi2 = K.hom([phi1((gamma^(-2))(z9))])
sage: L1.is_isomorphic_relative(L1cyc, base_isom=phi2)
False
sage: L2.is_isomorphic_relative(L1cyc, base_isom=phi2)
True
Omitting base_isom raises a ValueError when the base fields are not identical:
sage: L1.is_isomorphic_relative(L1cyc)
Traceback (most recent call last):
...
ValueError: other does not have the same base field as self, so an
˓→isomorphism from self's base_field to other's base_field must be provided
˓→using the base_isom parameter.
The parameter base_isom can also be used to check if the relative extensions are Galois conjugate:
sage: for g in G:
....:
if L1.is_isomorphic_relative(L2, g.as_hom()):
....:
print(g.as_hom())
Ring endomorphism of Number Field in z9 with defining polynomial x^6 + x^3 + 1
Defn: z9 |--> z9^4
lift_to_base ( element)
Lift an element of this extension into the base field if possible, or raise a ValueError if it is not possible.
EXAMPLES:
sage: x = polygen(ZZ)
sage: K.<a> = NumberField(x^3 - 2)
sage: R.<y> = K[]
sage: L.<b> = K.extension(y^2 - a)
sage: L.lift_to_base(b^4)
a^2
sage: L.lift_to_base(b^6)
2
sage: L.lift_to_base(355/113)
355/113
sage: L.lift_to_base(b)
Traceback (most recent call last):
...
ValueError: The element b is not in the base field
maximal_order ( v=None)
Return the maximal order, i.e., the ring of integers of this number field.
INPUT:
•v - (default: None) None, a prime, or a list of primes.
–if v is None, return the maximal order.
–if v is a prime, return an order that is p-maximal.
–if v is a list, return an order that is maximal at each prime in the list v.
EXAMPLES:
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sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order(); OK.basis()
[1, 1/2*a - 1/2*b, -1/2*b*a + 1/2, a]
sage: charpoly(OK.1)
x^2 + b*x + 1
sage: charpoly(OK.2)
x^2 - x + 1
sage: O2 = K.order([3*a, 2*b])
sage: O2.index_in(OK)
144
The following was previously “ridiculously slow”; see trac ticket #4738:
sage: K.<a,b> = NumberField([x^4 + 1, x^4 - 3])
sage: K.maximal_order()
Maximal Relative Order in Number Field in a with defining polynomial x^4 + 1
˓→over its base field
An example with nontrivial v :
sage: L.<a,b> = NumberField([x^2 - 3, x^2 - 5*49])
sage: O3 = L.maximal_order([3])
sage: O3.absolute_discriminant()
8643600
sage: O3.is_maximal()
False
ngens ( )
Return the number of generators of this relative number field.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]); K
Number Field in a with defining polynomial x^4 + 3 over its base field
sage: K.gens()
(a, b)
sage: K.ngens()
2
number_of_roots_of_unity ( )
Return number of roots of unity in this relative field.
EXAMPLES:
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
sage: K.number_of_roots_of_unity()
24
order ( *gens, **kwds)
Return the order with given ring generators in the maximal order of this number field.
INPUT:
•gens – list of elements of self; if no generators are given, just returns the cardinality of this number
field (oo) for consistency.
•check_is_integral – bool (default: True), whether to check that each generator is integral.
•check_rank – bool (default: True), whether to check that the ring generated by gens is of full rank.
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•allow_subfield – bool (default: False), if True and the generators do not generate an order, i.e.,
they generate a subring of smaller rank, instead of raising an error, return an order in a smaller number
field.
The check_is_integral and check_rank inputs must be given as explicit keyword arguments.
EXAMPLES:
sage: P.<a,b,c> = QQ[2^(1/2), 2^(1/3), 3^(1/2)]
sage: R = P.order([a,b,c]); R
Relative Order in Number Field in sqrt2 with defining polynomial x^2 - 2 over
˓→its base field
The base ring of an order in a relative extension is still Z.:
sage: R.base_ring()
Integer Ring
One must give enough generators to generate a ring of finite index in the maximal order:
sage: P.order([a,b])
Traceback (most recent call last):
...
ValueError: the rank of the span of gens is wrong
pari_absolute_base_polynomial ( )
Return the PARI polynomial defining the absolute base field, in y .
EXAMPLES:
sage: x = polygen(ZZ)
sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 3]); K
Number Field in a with defining polynomial x^2 + 2 over its base field
sage: K.pari_absolute_base_polynomial()
y^2 + 3
sage: type(K.pari_absolute_base_polynomial())
<type 'sage.libs.cypari2.gen.Gen'>
sage: z = ZZ['z'].0
sage: K.<a, b, c> = NumberField([z^2 + 2, z^2 + 3, z^2 + 5]); K
Number Field in a with defining polynomial z^2 + 2 over its base field
sage: K.pari_absolute_base_polynomial()
y^4 + 16*y^2 + 4
sage: K.base_field()
Number Field in b with defining polynomial z^2 + 3 over its base field
sage: len(QQ['y'](K.pari_absolute_base_polynomial()).roots(K.base_field()))
4
sage: type(K.pari_absolute_base_polynomial())
<type 'sage.libs.cypari2.gen.Gen'>
pari_relative_polynomial ( )
Return the PARI relative polynomial associated to this number field.
This is always a polynomial in x and y, suitable for PARI’s rnfinit function. Notice that if this is a relative
extension of a relative extension, the base field is the absolute base field.
EXAMPLES:
sage: k.<i> = NumberField(x^2 + 1)
sage: m.<z> = k.extension(k['w']([i,0,1]))
sage: m
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Number Field in z with defining polynomial w^2 + i over its base field
sage: m.pari_relative_polynomial()
Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)
sage: l.<t> = m.extension(m['t'].0^2 + z)
sage: l.pari_relative_polynomial()
Mod(1, y^4 + 1)*x^2 + Mod(y, y^4 + 1)
pari_rnf ( )
Return the PARI relative number field object associated to this relative extension.
EXAMPLES:
sage: k.<a> = NumberField([x^4 + 3, x^2 + 2])
sage: k.pari_rnf()
[x^4 + 3, [[364, -10*x^7 - 87*x^5 - 370*x^3 - 41*x], 1/364], [[108, 0; 0,
˓→108], 3], ...]
places ( all_complex=False, prec=None)
Return the collection of all infinite places of self.
By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one
of each pair of complex conjugate homomorphisms into CIF.
On the other hand, if prec is not None, we simply return places into RealField(prec) and ComplexField(prec) (or RDF, CDF if prec=53).
There is an optional flag all_complex, which defaults to False. If all_complex is True, then the real
embeddings are returned as embeddings into CIF instead of RIF.
EXAMPLES:
sage: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3])
sage: L.places()
[Relative number field morphism:
From: Number Field in b with defining polynomial x^2 - 5 over
To:
Real Field with 106 bits of precision
Defn: b |--> -2.236067977499789696409173668937
c |--> -1.213411662762229634132131377426,
Relative number field morphism:
From: Number Field in b with defining polynomial x^2 - 5 over
To:
Real Field with 106 bits of precision
Defn: b |--> 2.236067977499789696411548005367
c |--> -1.213411662762229634130492421800,
Relative number field morphism:
From: Number Field in b with defining polynomial x^2 - 5 over
To:
Complex Field with 53 bits of precision
Defn: b |--> -2.23606797749979 ...e-1...*I
c |--> 0.606705831381... - 1.45061224918844*I,
Relative number field morphism:
From: Number Field in b with defining polynomial x^2 - 5 over
To:
Complex Field with 53 bits of precision
Defn: b |--> 2.23606797749979 - 4.44089209850063e-16*I
c |--> 0.606705831381115 - 1.45061224918844*I]
its base field
its base field
its base field
its base field
polynomial ( )
For a relative number field, polynomial()
is deliberately not implemented.
relative_polynomial() or absolute_polynomial() must be used.
Either
EXAMPLES:
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sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
sage: K.polynomial()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field L you must use either L.
˓→relative_polynomial() or L.absolute_polynomial() as appropriate
relative_degree ( )
Returns the relative degree of this relative number field.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
sage: K.relative_degree()
2
relative_different ( )
Return the relative different of this extension 𝐿/𝐾 as an ideal of 𝐿. If you want the absolute different of
𝐿/Q, use L.absolute_different() .
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: PK.<t> = K[]
sage: L.<a> = K.extension(t^4 - i)
sage: L.relative_different()
Fractional ideal (4)
relative_discriminant ( )
Return the relative discriminant of this extension 𝐿/𝐾 as an ideal of 𝐾. If you want the (rational) discriminant of 𝐿/Q, use e.g. L.absolute_discriminant() .
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: t = K['t'].gen()
sage: L.<b> = K.extension(t^4 - i)
sage: L.relative_discriminant()
Fractional ideal (256)
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.relative_discriminant() == F.ideal(4*b)
True
relative_polynomial ( )
Return the defining polynomial of this relative number field over its base field.
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 + x + 1, x^3 + x + 1])
sage: K.relative_polynomial()
x^2 + x + 1
Use absolute polynomial for a polynomial that defines the absolute extension.:
sage: K.absolute_polynomial()
x^6 + 3*x^5 + 8*x^4 + 9*x^3 + 7*x^2 + 6*x + 3
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relative_vector_space ( )
Return vector space over the base field of self and isomorphisms from the vector space to self and in the
other direction.
EXAMPLES:
sage: K.<a,b,c> = NumberField([x^2 + 2, x^3 + 2, x^3 + 3]); K
Number Field in a with defining polynomial x^2 + 2 over its base field
sage: V, from_V, to_V = K.relative_vector_space()
sage: from_V(V.0)
1
sage: to_V(K.0)
(0, 1)
sage: from_V(to_V(K.0))
a
sage: to_V(from_V(V.0))
(1, 0)
sage: to_V(from_V(V.1))
(0, 1)
The underlying vector space and maps is cached:
sage: W, from_V, to_V = K.relative_vector_space()
sage: V is W
True
relativize ( alpha, names)
Given an element in self or an embedding of a subfield into self, return a relative number field 𝐾 isomorphic to self that is relative over the absolute field Q(𝛼) or the domain of 𝛼, along with isomorphisms from
𝐾 to self and from self to 𝐾.
INPUT:
•alpha – an element of self, or an embedding of a subfield into self
•names – name of generator for output field 𝐾.
OUTPUT: 𝐾 – a relative number field
Also, K.structure() returns from_K and to_K , where from_K is an isomorphism from 𝐾 to
self and to_K is an isomorphism from self to 𝐾.
EXAMPLES:
sage: K.<a,b> = NumberField([x^4 + 3, x^2 + 2]);
Number Field in a with defining polynomial x^4 +
sage: L.<z,w> = K.relativize(a^2)
sage: z^2
z^2
sage: w^2
-3
sage: L
Number Field in z with defining polynomial x^4 +
˓→over its base field
sage: L.base_field()
Number Field in w with defining polynomial x^2 +
K
3 over its base field
(-2*w + 4)*x^2 + 4*w + 1
3
Now suppose we have 𝐾 below 𝐿 below 𝑀 :
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sage: M = NumberField(x^8 + 2, 'a'); M
Number Field in a with defining polynomial x^8 + 2
sage: L, L_into_M, _ = M.subfields(4)[0]; L
Number Field in a0 with defining polynomial x^4 + 2
sage: K, K_into_L, _ = L.subfields(2)[0]; K
Number Field in a0_0 with defining polynomial x^2 + 2
sage: K_into_M = L_into_M * K_into_L
sage: L_over_K = L.relativize(K_into_L, 'c'); L_over_K
Number Field in c with defining polynomial x^2 + a0_0 over its base field
sage: L_over_K_to_L, L_to_L_over_K = L_over_K.structure()
sage: M_over_L_over_K = M.relativize(L_into_M * L_over_K_to_L, 'd'); M_over_L_
˓→over_K
Number Field in d with defining polynomial x^2 + c over its base field
sage: M_over_L_over_K.base_field() is L_over_K
True
Test relativizing a degree 6 field over its degree 2 and degree 3 subfields, using both an explicit element:
sage: K.<a> = NumberField(x^6 + 2); K
Number Field in a with defining polynomial x^6 + 2
sage: K2, K2_into_K, _ = K.subfields(2)[0]; K2
Number Field in a0 with defining polynomial x^2 + 2
sage: K3, K3_into_K, _ = K.subfields(3)[0]; K3
Number Field in a0 with defining polynomial x^3 - 2
Here we explicitly relativize over an element of K2 (not the generator):
sage: L = K.relativize(K3_into_K, 'b'); L
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: L_to_K, K_to_L = L.structure()
sage: L_over_K2 = L.relativize(K_to_L(K2_into_K(K2.gen() + 1)), 'c'); L_over_
˓→K2
Number Field in c0 with defining polynomial x^3 - c1 + 1 over its base field
sage: L_over_K2.base_field()
Number Field in c1 with defining polynomial x^2 - 2*x + 3
Here we use a morphism to preserve the base field information:
sage: K2_into_L = K_to_L * K2_into_K
sage: L_over_K2 = L.relativize(K2_into_L, 'c'); L_over_K2
Number Field in c with defining polynomial x^3 - a0 over its base field
sage: L_over_K2.base_field() is K2
True
roots_of_unity ( )
Return all the roots of unity in this relative field, primitive or not.
EXAMPLES:
sage: K.<a, b> = NumberField( [x^2 + x + 1, x^4 + 1] )
sage: K.roots_of_unity()[:5]
[b*a + b, b^2*a, -b^3, a + 1, b*a]
subfields ( degree=0, name=None)
Return all subfields of this relative number field self of the given degree, or of all possible degrees if degree
is 0. The subfields are returned as absolute fields together with an embedding into self. For the case of the
field itself, the reverse isomorphism is also provided.
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EXAMPLES:
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.subfields(2)
[
(Number Field in c0 with defining polynomial x^2 - 24*x + 72, Ring morphism:
From: Number Field in c0 with defining polynomial x^2 - 24*x + 72
To:
Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b ˓→6 over its base field
Defn: c0 |--> -6*a + 12, None),
(Number Field in c1 with defining polynomial x^2 - 24*x + 120, Ring morphism:
From: Number Field in c1 with defining polynomial x^2 - 24*x + 120
To:
Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b ˓→6 over its base field
Defn: c1 |--> 2*b*a + 12, None),
(Number Field in c2 with defining polynomial x^2 - 24*x + 96, Ring morphism:
From: Number Field in c2 with defining polynomial x^2 - 24*x + 96
To:
Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b ˓→6 over its base field
Defn: c2 |--> -4*b + 12, None)
]
sage: K.subfields(8, 'w')
[
(Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 - 36*x^2 +
˓→9, Ring morphism:
From: Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 ˓→36*x^2 + 9
To:
Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b ˓→6 over its base field
Defn: w0 |--> (-1/2*b*a + 1/2*b + 1/2)*c, Relative number field morphism:
From: Number Field in c with defining polynomial Y^2 + (-2*b - 3)*a - 2*b ˓→6 over its base field
To:
Number Field in w0 with defining polynomial x^8 - 12*x^6 + 36*x^4 ˓→36*x^2 + 9
Defn: c |--> -1/3*w0^7 + 4*w0^5 - 12*w0^3 + 11*w0
a |--> 1/3*w0^6 - 10/3*w0^4 + 5*w0^2
b |--> -2/3*w0^6 + 7*w0^4 - 14*w0^2 + 6)
]
sage: K.subfields(3)
[]
uniformizer ( P, others=’positive’)
Returns an element of self with valuation 1 at the prime ideal P.
INPUT:
•self - a number field
•P - a prime ideal of self
•others - either “positive” (default), in which case the element will have non-negative valuation at
all other primes of self, or “negative”, in which case the element will have non-positive valuation at
all other primes of self.
Note: When P is principal (e.g. always when self has class number one) the result may or may not be a
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generator of P!
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 3])
sage: P = K.prime_factors(5)[0]; P
Fractional ideal (5, 1/2*a - b - 5/2)
sage: u = K.uniformizer(P)
sage: u.valuation(P)
1
sage: (P, 1) in K.factor(u)
True
vector_space ( )
For a relative number field, vector_space() is deliberately not implemented, so that a user cannot
confuse relative_vector_space() with absolute_vector_space() .
EXAMPLES:
sage: K.<a> = NumberFieldTower([x^2 - 17, x^3 - 2])
sage: K.vector_space()
Traceback (most recent call last):
...
NotImplementedError: For a relative number field L you must use either L.
˓→relative_vector_space() or L.absolute_vector_space() as appropriate
sage.rings.number_field.number_field_rel. NumberField_relative_v1 ( base_field,
poly,
name, latex_name,
canonical_embedding=None)
Used for unpickling old pickles.
EXAMPLES:
sage: from sage.rings.number_field.number_field_rel import NumberField_relative_v1
sage: R.<x> = CyclotomicField(3)[]
sage: NumberField_relative_v1(CyclotomicField(3), x^2 + 7, 'a', 'a')
Number Field in a with defining polynomial x^2 + 7 over its base field
sage.rings.number_field.number_field_rel. is_RelativeNumberField ( x)
Return True if 𝑥 is a relative number field.
EXAMPLES:
sage: from sage.rings.number_field.number_field_rel import is_RelativeNumberField
sage: is_RelativeNumberField(NumberField(x^2+1,'a'))
False
sage: k.<a> = NumberField(x^3 - 2)
sage: l.<b> = k.extension(x^3 - 3); l
Number Field in b with defining polynomial x^3 - 3 over its base field
sage: is_RelativeNumberField(l)
True
sage: is_RelativeNumberField(QQ)
False
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1.4 Number Field Elements
AUTHORS:
• William Stein: version before it got Cython’d
• Joel B. Mohler (2007-03-09): First reimplementation in Cython
• William Stein (2007-09-04): add doctests
• Robert Bradshaw (2007-09-15): specialized classes for relative and absolute elements
• John Cremona (2009-05-15): added support for local and global logarithmic heights.
• Robert Harron (2012-08): conjugate() now works for all fields contained in CM fields
class sage.rings.number_field.number_field_element. CoordinateFunction ( alpha,
W,
to_V)
This class provides a callable object which expresses elements in terms of powers of a fixed field generator 𝛼.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + x + 3)
sage: f = (a + 1).coordinates_in_terms_of_powers(); f
Coordinate function that writes elements in terms of the powers of a + 1
sage: f.__class__
<class sage.rings.number_field.number_field_element.CoordinateFunction at ...>
sage: f(a)
[-1, 1]
sage: f == loads(dumps(f))
True
alpha ( )
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 2)
sage: (a + 2).coordinates_in_terms_of_powers().alpha()
a + 2
class sage.rings.number_field.number_field_element. NumberFieldElement
Bases: sage.structure.element.FieldElement
An element of a number field.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + x + 1)
sage: a^3
-a - 1
abs ( prec=None, i=None)
Return the absolute value of this element.
If i is provided, then the absolute value of the 𝑖-th embedding is given.
Otherwise, if the number field has a coercion embedding into R, the corresponding absolute value is
returned as an element of the same field (unless prec is given). Otherwise, if it has a coercion embedding
into C, then the corresponding absolute value is returned. Finally, if there is no coercion embedding, 𝑖
defaults to 0.
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For the computation, the complex field with prec bits of precision is used, defaulting to 53 bits of
precision if prec is not provided. The result is in the corresponding real field.
INPUT:
•prec - (default: None) integer bits of precision
•i - (default: None) integer, which embedding to use
EXAMPLES:
sage: z = CyclotomicField(7).gen()
sage: abs(z)
1.00000000000000
sage: abs(z^2 + 17*z - 3)
16.0604426799931
sage: K.<a> = NumberField(x^3+17)
sage: abs(a)
2.57128159065824
sage: a.abs(prec=100)
2.5712815906582353554531872087
sage: a.abs(prec=100,i=1)
2.5712815906582353554531872087
sage: a.abs(100, 2)
2.5712815906582353554531872087
Here’s one where the absolute value depends on the embedding:
sage: K.<b> = NumberField(x^2-2)
sage: a = 1 + b
sage: a.abs(i=0)
0.414213562373095
sage: a.abs(i=1)
2.41421356237309
Check that trac ticket #16147 is fixed:
sage: x = polygen(ZZ)
sage: f = x^3 - x - 1
sage: beta = f.complex_roots()[0]; beta
1.32471795724475
sage: K.<b> = NumberField(f, embedding=beta)
sage: b.abs()
1.32471795724475
Check that for fields with real coercion embeddings, absolute values are in the same field (trac ticket
#21105):
sage:
sage:
sage:
sage:
b
x = polygen(ZZ)
f = x^3 - x - 1
K.<b> = NumberField(f, embedding=1.3)
b.abs()
However, if a specific embedding is requested, the behavior reverts to that of number fields without a
coercion embedding into R:
sage: b.abs(i=2)
1.32471795724475
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Also, if a precision is requested explicitly, the behavior reverts to that of number fields without a coercion
embedding into R:
sage: b.abs(prec=53)
1.32471795724475
abs_non_arch ( P, prec=None)
Return the non-archimedean absolute value of this element with respect to the prime 𝑃 , to the given
precision.
INPUT:
•P - a prime ideal of the parent of self
•prec (int) – desired floating point precision (default: default RealField precision).
OUTPUT:
(real) the non-archimedean absolute value of this element with respect to the prime 𝑃 , to the given precision. This is the normalised absolute value, so that the underlying prime number 𝑝 has absolute value
1/𝑝.
EXAMPLES:
sage: K.<a> = NumberField(x^2+5)
sage: [1/K(2).abs_non_arch(P) for P in K.primes_above(2)]
[2.00000000000000]
sage: [1/K(3).abs_non_arch(P) for P in K.primes_above(3)]
[3.00000000000000, 3.00000000000000]
sage: [1/K(5).abs_non_arch(P) for P in K.primes_above(5)]
[5.00000000000000]
A relative example:
sage: L.<b> = K.extension(x^2-5)
sage: [b.abs_non_arch(P) for P in L.primes_above(b)]
[0.447213595499958, 0.447213595499958]
absolute_norm ( )
Return the absolute norm of this number field element.
EXAMPLES:
sage: K1.<a1> = CyclotomicField(11)
sage: K2.<a2> = K1.extension(x^2 - 3)
sage: K3.<a3> = K2.extension(x^2 + 1)
sage: (a1 + a2 + a3).absolute_norm()
1353244757701
sage: QQ(7/5).absolute_norm()
7/5
additive_order ( )
Return the additive order of this element (i.e. infinity if self != 0, 1 if self == 0)
EXAMPLES:
sage: K.<u> = NumberField(x^4 - 3*x^2 + 3)
sage: u.additive_order()
+Infinity
sage: K(0).additive_order()
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1
sage: K.ring_of_integers().characteristic() # implicit doctest
0
ceil ( )
Return the ceiling of this number field element.
EXAMPLES:
sage: x = polygen(ZZ)
sage: p = x**7 - 5*x**2 + x + 1
sage: a_AA = AA.polynomial_root(p, RIF(1,2))
sage: K.<a> = NumberField(p, embedding=a_AA)
sage: b = a**5 + a/2 - 1/7
sage: RR(b)
4.13444473767055
sage: b.ceil()
5
This function always succeeds even if a tremendous precision is needed:
sage: c = b - 5065701199253/1225243417356 + 2
sage: c.ceil()
3
sage: RIF(c).unique_ceil()
Traceback (most recent call last):
...
ValueError: interval does not have a unique ceil
If the number field is not embedded, this function is valid only if the element is rational:
sage: p = x**5 - 3
sage: K.<a> = NumberField(p)
sage: K(2/3).ceil()
1
sage: a.ceil()
Traceback (most recent call last):
...
TypeError: ceil not uniquely defined since no real embedding is specified
charpoly ( var=’x’)
Return the characteristic polynomial of this number field element.
EXAMPLES:
sage:
sage:
x^3 +
sage:
x^3 -
K.<a> = NumberField(x^3 + 7)
a.charpoly()
7
K(1).charpoly()
3*x^2 + 3*x - 1
complex_embedding ( prec=53, i=0)
Return the i-th embedding of self in the complex numbers, to the given precision.
EXAMPLES:
sage: k.<a> = NumberField(x^3 - 2)
sage: a.complex_embedding()
-0.629960524947437 - 1.09112363597172*I
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sage: a.complex_embedding(10)
-0.63 - 1.1*I
sage: a.complex_embedding(100)
-0.62996052494743658238360530364 - 1.0911236359717214035600726142*I
sage: a.complex_embedding(20, 1)
-0.62996 + 1.0911*I
sage: a.complex_embedding(20, 2)
1.2599
complex_embeddings ( prec=53)
Return the images of this element in the floating point complex numbers, to the given bits of precision.
INPUT:
•prec - integer (default: 53) bits of precision
EXAMPLES:
sage: k.<a> = NumberField(x^3 - 2)
sage: a.complex_embeddings()
[-0.629960524947437 - 1.09112363597172*I, -0.629960524947437 + 1.
˓→09112363597172*I, 1.25992104989487]
sage: a.complex_embeddings(10)
[-0.63 - 1.1*I, -0.63 + 1.1*I, 1.3]
sage: a.complex_embeddings(100)
[-0.62996052494743658238360530364 - 1.0911236359717214035600726142*I, -0.
˓→62996052494743658238360530364 + 1.0911236359717214035600726142*I, 1.
˓→2599210498948731647672106073]
conjugate ( )
Return the complex conjugate of the number field element.
This is only well-defined for fields contained in CM fields (i.e. for totally real fields and CM fields).
Recall that a CM field is a totally imaginary quadratic extension of a totally real field. For other fields, a
ValueError is raised.
EXAMPLES:
sage: k.<I> = QuadraticField(-1)
sage: I.conjugate()
-I
sage: (I/(1+I)).conjugate()
-1/2*I + 1/2
sage: z6 = CyclotomicField(6).gen(0)
sage: (2*z6).conjugate()
-2*zeta6 + 2
The following example now works.
sage: F.<b> = NumberField(x^2 - 2)
sage: K.<j> = F.extension(x^2 + 1)
sage: j.conjugate()
-j
Raise a ValueError if the field is not contained in a CM field.
sage: K.<b> = NumberField(x^3 - 2)
sage: b.conjugate()
Traceback (most recent call last):
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...
ValueError: Complex conjugation is only well-defined for fields contained in
˓→CM fields.
An example of a non-quadratic totally real field.
sage: F.<a> = NumberField(x^4 + x^3 - 3*x^2 - x + 1)
sage: a.conjugate()
a
An example of a non-cyclotomic CM field.
sage: K.<a> = NumberField(x^4 - x^3 + 2*x^2 + x + 1)
sage: a.conjugate()
-1/2*a^3 - a - 1/2
sage: (2*a^2 - 1).conjugate()
a^3 - 2*a^2 - 2
coordinates_in_terms_of_powers ( )
Let 𝛼 be self. Return a callable object (of type CoordinateFunction ) that takes any element of the
parent of self in Q(𝛼) and writes it in terms of the powers of 𝛼: 1, 𝛼, 𝛼2 , ....
(NOT CACHED).
EXAMPLES:
This function allows us to write elements of a number field in terms of a different generator without having
to construct a whole separate number field.
sage: y = polygen(QQ,'y'); K.<beta> = NumberField(y^3 - 2); K
Number Field in beta with defining polynomial y^3 - 2
sage: alpha = beta^2 + beta + 1
sage: c = alpha.coordinates_in_terms_of_powers(); c
Coordinate function that writes elements in terms of the powers of beta^2 +
˓→beta + 1
sage: c(beta)
[-2, -3, 1]
sage: c(alpha)
[0, 1, 0]
sage: c((1+beta)^5)
[3, 3, 3]
sage: c((1+beta)^10)
[54, 162, 189]
This function works even if self only generates a subfield of this number field.
sage: k.<a> = NumberField(x^6 - 5)
sage: alpha = a^3
sage: c = alpha.coordinates_in_terms_of_powers()
sage: c((2/3)*a^3 - 5/3)
[-5/3, 2/3]
sage: c
Coordinate function that writes elements in terms of the powers of a^3
sage: c(a)
Traceback (most recent call last):
...
ArithmeticError: vector is not in free module
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denominator ( )
Return the denominator of this element, which is by definition the denominator of the corresponding
polynomial representation. I.e., elements of number fields are represented as a polynomial (in reduced
form) modulo the modulus of the number field, and the denominator is the denominator of this polynomial.
EXAMPLES:
sage: K.<z> = CyclotomicField(3)
sage: a = 1/3 + (1/5)*z
sage: a.denominator()
15
denominator_ideal ( )
Return the denominator ideal of this number field element.
The denominator ideal of a number field element 𝑎 is the integral ideal consisting of all elements of the
ring of integers 𝑅 whose product with 𝑎 is also in 𝑅.
See also:
numerator_ideal()
EXAMPLES:
sage: K.<a> = NumberField(x^2+5)
sage: b = (1+a)/2
sage: b.norm()
3/2
sage: D = b.denominator_ideal(); D
Fractional ideal (2, a + 1)
sage: D.norm()
2
sage: (1/b).denominator_ideal()
Fractional ideal (3, a + 1)
sage: K(0).denominator_ideal()
Fractional ideal (1)
descend_mod_power ( K=’QQ’, d=2)
Return a list of elements of the subfield 𝐾 equal to self modulo 𝑑‘th powers.
INPUT:
•K (number field, default QQ) – a subfield of the parent number field 𝐿 of self
•d (positive integer, default 2) – an integer at least 2
OUTPUT:
A list, possibly empty, of elements of K equal to self modulo 𝑑‘th powers, i.e. the preimages of self
under the map 𝐾 * /(𝐾 * )𝑑 → 𝐿* /(𝐿* )𝑑 where 𝐿 is the parent of self . A ValueError is raised if 𝐾
does not embed into 𝐿.
ALGORITHM:
All preimages must lie in the Selmer group 𝐾(𝑆, 𝑑) for a suitable finite set of primes 𝑆, which reduces the
question to a finite set of possibilities. We may take 𝑆 to be the set of primes which ramify in 𝐿 together
with those for which the valuation of self is not divisible by 𝑑.
EXAMPLES:
A relative example:
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sage:
sage:
sage:
sage:
sage:
[-3*i
Qi.<i> = QuadraticField(-1)
K.<zeta> = CyclotomicField(8)
f = Qi.embeddings(K)[0]
a = f(2+3*i) * (2-zeta)^2
a.descend_mod_power(Qi,2)
- 2, -2*i + 3]
An absolute example:
sage: K.<zeta> = CyclotomicField(8)
sage: K(1).descend_mod_power(QQ,2)
[1, 2, -1, -2]
sage: a = 17*K.random_element()^2
sage: a.descend_mod_power(QQ,2)
[17, 34, -17, -34]
factor ( )
Return factorization of this element into prime elements and a unit.
OUTPUT:
(Factorization) If all the prime ideals in the support are principal, the output is a Factorization as a product
of prime elements raised to appropriate powers, with an appropriate unit factor.
Raise ValueError if the factorization of the ideal (self) contains a non-principal prime ideal.
EXAMPLES:
sage: K.<i> = NumberField(x^2+1)
sage: (6*i + 6).factor()
(-i) * (i + 1)^3 * 3
In the following example, the class number is 2. If a factorization in prime elements exists, we will find it:
sage: K.<a> = NumberField(x^2-10)
sage: factor(169*a + 531)
(-6*a - 19) * (-3*a - 1) * (-2*a + 9)
sage: factor(K(3))
Traceback (most recent call last):
...
ArithmeticError: non-principal ideal in factorization
Factorization of 0 is not allowed:
sage: K.<i> = QuadraticField(-1)
sage: K(0).factor()
Traceback (most recent call last):
...
ArithmeticError: factorization of 0 is not defined
floor ( )
Return the floor of this number field element.
EXAMPLES:
sage:
sage:
sage:
sage:
124
x = polygen(ZZ)
p = x**7 - 5*x**2 + x + 1
a_AA = AA.polynomial_root(p, RIF(1,2))
K.<a> = NumberField(p, embedding=a_AA)
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sage: b = a**5 + a/2 - 1/7
sage: RR(b)
4.13444473767055
sage: b.floor()
4
This function always succeeds even if a tremendous precision is needed:
sage: c = b - 4772404052447/1154303505127 + 2
sage: c.floor()
1
sage: RIF(c).unique_floor()
Traceback (most recent call last):
...
ValueError: interval does not have a unique floor
If the number field is not embedded, this function is valid only if the element is rational:
sage: p = x**5 - 3
sage: K.<a> = NumberField(p)
sage: K(2/3).floor()
0
sage: a.floor()
Traceback (most recent call last):
...
TypeError: floor not uniquely defined since no real embedding is specified
galois_conjugates ( K)
Return all Gal(Qbar/Q)-conjugates of this number field element in the field K.
EXAMPLES:
In the first example the conjugates are obvious:
sage: K.<a> = NumberField(x^2 - 2)
sage: a.galois_conjugates(K)
[a, -a]
sage: K(3).galois_conjugates(K)
[3]
In this example the field is not Galois, so we have to pass to an extension to obtain the Galois conjugates.
sage: K.<a> = NumberField(x^3 - 2)
sage: c = a.galois_conjugates(K); c
[a]
sage: K.<a> = NumberField(x^3 - 2)
sage: c = a.galois_conjugates(K.galois_closure('a1')); c
[1/18*a1^4, -1/36*a1^4 + 1/2*a1, -1/36*a1^4 - 1/2*a1]
sage: c[0]^3
2
sage: parent(c[0])
Number Field in a1 with defining polynomial x^6 + 108
sage: parent(c[0]).is_galois()
True
There is only one Galois conjugate of
1.4. Number Field Elements
√
3
√
2 in Q( 3 2).
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sage: a.galois_conjugates(K)
[a]
Galois conjugates of
√
3
2 in the field Q(𝜁3 ,
√
3
2):
sage: L.<a> = CyclotomicField(3).extension(x^3 - 2)
sage: a.galois_conjugates(L)
[a, (-zeta3 - 1)*a, zeta3*a]
gcd ( other)
Return the greatest common divisor of self and other .
INPUT:
•self , other – elements of a number field or maximal order.
OUTPUT:
•A generator of the ideal (self,other) . If the parent is a number field, this always returns 0 or 1.
For maximal orders, this raises ArithmeticError if the ideal is not principal.
EXAMPLES:
sage:
sage:
1
sage:
1
sage:
0
sage:
sage:
i + 1
K.<i> = QuadraticField(-1)
(i+1).gcd(2)
K(1).gcd(0)
K(0).gcd(0)
R = K.maximal_order()
R(i+1).gcd(2)
Non-maximal orders are not supported:
sage: R = K.order(2*i)
sage: R(1).gcd(R(4*i))
Traceback (most recent call last):
...
NotImplementedError: gcd() for Order in Number Field in i with defining
˓→polynomial x^2 + 1 is not implemented
The following field has class number 3, but if the ideal (self,other) happens to be principal, this still
works:
sage: K.<a> = NumberField(x^3 - 7)
sage: K.class_number()
3
sage: a.gcd(7)
1
sage: R = K.maximal_order()
sage: R(a).gcd(7)
a
sage: R(a+1).gcd(2)
Traceback (most recent call last):
...
ArithmeticError: ideal (a + 1, 2) is not principal, gcd is not defined
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sage: R(2*a - a^2).gcd(0)
a
global_height ( prec=None)
Returns the absolute logarithmic height of this number field element.
INPUT:
•prec (int) – desired floating point precision (default: default RealField precision).
OUTPUT:
(real) The absolute logarithmic height of this number field element; that is, the sum of the local heights at
all finite and infinite places, scaled by the degree to make the result independent of the parent field.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: b = a/2
sage: b.global_height()
0.789780699008...
sage: b.global_height(prec=200)
0.78978069900813892060267152032141577237037181070060784564457
The global height of an algebraic number is absolute, i.e. it does not depend on the parent field:
sage: QQ(6).global_height()
1.79175946922805
sage: K(6).global_height()
1.79175946922805
sage: L.<b> = NumberField((a^2).minpoly())
sage: L.degree()
2
sage: b.global_height() # element of L (degree 2 field)
1.41660667202811
sage: (a^2).global_height() # element of K (degree 4 field)
1.41660667202811
And of course every element has the same height as it’s inverse:
sage: K.<s> = QuadraticField(2)
sage: s.global_height()
0.346573590279973
sage: (1/s).global_height()
#make sure that 11758 is fixed
0.346573590279973
global_height_arch ( prec=None)
Returns the total archimedean component of the height of self.
INPUT:
•prec (int) – desired floating point precision (default: default RealField precision).
OUTPUT:
(real) The total archimedean component of the height of this number field element; that is, the sum of the
local heights at all infinite places.
EXAMPLES:
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sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: b = a/2
sage: b.global_height_arch()
0.38653407379277...
global_height_non_arch ( prec=None)
Returns the total non-archimedean component of the height of self.
INPUT:
•prec (int) – desired floating point precision (default: default RealField precision).
OUTPUT:
(real) The total non-archimedean component of the height of this number field element; that is, the sum of
the local heights at all finite places, weighted by the local degrees.
ALGORITHM:
An alternative formula is log(𝑑) where 𝑑 is the norm of the denominator ideal; this is used to avoid
factorization.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: b = a/6
sage: b.global_height_non_arch()
7.16703787691222
Check that this is equal to the sum of the non-archimedean local heights:
sage: [b.local_height(P) for P in b.support()]
[0.000000000000000, 0.693147180559945, 1.09861228866811, 1.09861228866811]
sage: [b.local_height(P, weighted=True) for P in b.support()]
[0.000000000000000, 2.77258872223978, 2.19722457733622, 2.19722457733622]
sage: sum([b.local_height(P,weighted=True) for P in b.support()])
7.16703787691222
A relative example:
sage: PK.<y> = K[]
sage: L.<c> = NumberField(y^2 + a)
sage: (c/10).global_height_non_arch()
18.4206807439524
inverse_mod ( I)
Returns the inverse of self mod the integral ideal I.
INPUT:
•I - may be an ideal of self.parent(), or an element or list of elements of self.parent() generating a
nonzero ideal. A ValueError is raised if I is non-integral or zero. A ZeroDivisionError is raised if I +
(x) != (1).
NOTE: It’s not implemented yet for non-integral elements.
EXAMPLES:
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sage:
sage:
sage:
sage:
1
k.<a> = NumberField(x^2 + 23)
N = k.ideal(3)
d = 3*a + 1
d.inverse_mod(N)
sage:
sage:
sage:
True
sage:
True
sage:
sage:
7
sage:
True
sage:
True
k.<a> = NumberField(x^3 + 11)
d = a + 13
d.inverse_mod(a^2)*d - 1 in k.ideal(a^2)
d.inverse_mod((5, a + 1))*d - 1 in k.ideal(5, a + 1)
K.<b> = k.extension(x^2 + 3)
b.inverse_mod([37, a - b])
7*b - 1 in K.ideal(37, a - b)
b.inverse_mod([37, a - b]).parent() == K
is_integer ( )
Test whether this number field element is an integer
See also:
•is_rational() to test if this element is a rational number
•is_integral() to test if this element is an algebraic integer
EXAMPLES:
sage:
sage:
False
sage:
False
sage:
True
sage:
True
sage:
False
K.<cbrt3> = NumberField(x^3 - 3)
cbrt3.is_integer()
(cbrt3**2 - cbrt3 + 2).is_integer()
K(-12).is_integer()
K(0).is_integer()
K(1/2).is_integer()
is_integral ( )
Determine if a number is in the ring of integers of this number field.
EXAMPLES:
sage:
sage:
True
sage:
sage:
True
sage:
x^2 sage:
sage:
K.<a> = NumberField(x^2 + 23)
a.is_integral()
t = (1+a)/2
t.is_integral()
t.minpoly()
x + 6
t = a/2
t.is_integral()
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False
sage: t.minpoly()
x^2 + 23/4
An example in a relative extension:
sage: K.<a,b> = NumberField([x^2+1, x^2+3])
sage: (a+b).is_integral()
True
sage: ((a-b)/2).is_integral()
False
is_norm ( L, element=False, proof=True)
Determine whether self is the relative norm of an element of L/K, where K is self.parent().
INPUT:
•L – a number field containing K=self.parent()
•element – True or False, whether to also output an element of which self is a norm
•proof – If True, then the output is correct unconditionally. If False, then the output is correct under
GRH.
OUTPUT:
If element is False, then the output is a boolean B, which is True if and only if self is the relative norm of
an element of L to K. If element is False, then the output is a pair (B, x), where B is as above. If B is True,
then x is an element of L such that self == x.norm(K). Otherwise, x is None.
ALGORITHM:
Uses PARI’s rnfisnorm. See self._rnfisnorm().
EXAMPLES:
sage:
sage:
sage:
sage:
False
sage:
True
K.<beta> = NumberField(x^3+5)
Q.<X> = K[]
L = K.extension(X^2+X+beta, 'gamma')
(beta/2).is_norm(L)
beta.is_norm(L)
With a relative base field:
sage:
sage:
sage:
True
sage:
sage:
True
K.<a, b> = NumberField([x^2 - 2, x^2 - 3])
L.<c> = K.extension(x^2 - 5)
(2*a*b).is_norm(L)
_, v = (2*b*a).is_norm(L, element=True)
v.norm(K) == 2*a*b
Non-Galois number fields:
sage:
sage:
sage:
sage:
True
130
K.<a> = NumberField(x^2 + x + 1)
Q.<X> = K[]
L.<b> = NumberField(X^4 + a + 2)
(a/4).is_norm(L)
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sage: (a/2).is_norm(L)
Traceback (most recent call last):
...
NotImplementedError: is_norm is not implemented unconditionally for norms
˓→from non-Galois number fields
sage: (a/2).is_norm(L, proof=False)
False
sage: K.<a> = NumberField(x^3 + x + 1)
sage: Q.<X> = K[]
sage: L.<b> = NumberField(X^4 + a)
sage: t = (-a).is_norm(L, element=True); t
(True, b^3 + 1)
sage: t[1].norm(K)
-a
AUTHORS:
•Craig Citro (2008-04-05)
•Marco Streng (2010-12-03)
is_nth_power ( n)
Return True if self is an 𝑛‘th power in its parent 𝐾.
EXAMPLES:
sage:
sage:
True
sage:
True
sage:
False
sage:
True
K.<a> = NumberField(x^4-7)
K(7).is_nth_power(2)
K(7).is_nth_power(4)
K(7).is_nth_power(8)
K((a-3)^5).is_nth_power(5)
ALGORITHM: Use PARI to factor 𝑥𝑛 - self in 𝐾.
is_one ( )
Test whether this number field element is 1.
EXAMPLES:
sage:
sage:
True
sage:
False
sage:
False
sage:
False
sage:
False
K.<a> = NumberField(x^3 + 3)
K(1).is_one()
K(0).is_one()
K(-1).is_one()
K(1/2).is_one()
a.is_one()
is_rational ( )
Test whether this number field element is a rational number
See also:
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•is_integer() to test if this element is an integer
•is_integral() to test if this element is an algebraic integer
EXAMPLES:
sage:
sage:
False
sage:
False
sage:
True
sage:
True
sage:
True
K.<cbrt3> = NumberField(x^3 - 3)
cbrt3.is_rational()
(cbrt3**2 - cbrt3 + 1/2).is_rational()
K(-12).is_rational()
K(0).is_rational()
K(1/2).is_rational()
is_square ( root=False)
Return True if self is a square in its parent number field and otherwise return False.
INPUT:
•root - if True, also return a square root (or None if self is not a perfect square)
EXAMPLES:
sage: m.<b> = NumberField(x^4 - 1789)
sage: b.is_square()
False
sage: c = (2/3*b + 5)^2; c
4/9*b^2 + 20/3*b + 25
sage: c.is_square()
True
sage: c.is_square(True)
(True, 2/3*b + 5)
We also test the functional notation.
sage: is_square(c, True)
(True, 2/3*b + 5)
sage: is_square(c)
True
sage: is_square(c+1)
False
is_totally_positive ( )
Returns True if self is positive for all real embeddings of its parent number field. We do nothing at complex
places, so e.g. any element of a totally complex number field will return True.
EXAMPLES:
sage: F.<b> = NumberField(x^3-3*x-1)
sage: b.is_totally_positive()
False
sage: (b^2).is_totally_positive()
True
is_unit ( )
Return True if self is a unit in the ring where it is defined.
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EXAMPLES:
sage:
sage:
sage:
True
sage:
False
sage:
True
K.<a> = NumberField(x^2 - x - 1)
OK = K.ring_of_integers()
OK(a).is_unit()
OK(13).is_unit()
K(13).is_unit()
It also works for relative fields and orders:
sage:
sage:
sage:
True
sage:
False
sage:
True
K.<a,b> = NumberField([x^2 - 3, x^4 + x^3 + x^2 + x + 1])
OK = K.ring_of_integers()
OK(b).is_unit()
OK(a).is_unit()
a.is_unit()
list ( )
Return the list of coefficients of self written in terms of a power basis.
EXAMPLES:
sage: K.<a> = NumberField(x^3 - x + 2); ((a + 1)/(a + 2)).list()
[1/4, 1/2, -1/4]
sage: K.<a, b> = NumberField([x^3 - x + 2, x^2 + 23]); ((a + b)/(a + 2)).
˓→list()
[3/4*b - 1/2, -1/2*b + 1, 1/4*b - 1/2]
local_height ( P, prec=None, weighted=False)
Returns the local height of self at a given prime ideal 𝑃 .
INPUT:
•P - a prime ideal of the parent of self
•prec (int) – desired floating point precision (default: default RealField precision).
•weighted (bool, default False) – if True, apply local degree weighting.
OUTPUT:
(real) The local height of this number field element at the place 𝑃 . If weighted is True, this is multiplied
by the local degree (as required for global heights).
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: P = K.ideal(61).factor()[0][0]
sage: b = 1/(a^2 + 30)
sage: b.local_height(P)
4.11087386417331
sage: b.local_height(P, weighted=True)
8.22174772834662
sage: b.local_height(P, 200)
4.1108738641733112487513891034256147463156817430812610629374
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sage: (b^2).local_height(P)
8.22174772834662
sage: (b^-1).local_height(P)
0.000000000000000
A relative example:
sage: PK.<y> = K[]
sage: L.<c> = NumberField(y^2 + a)
sage: L(1/4).local_height(L.ideal(2, c-a+1))
1.38629436111989
local_height_arch ( i, prec=None, weighted=False)
Returns the local height of self at the 𝑖‘th infinite place.
INPUT:
•i (int) - an integer in range(r+s) where (𝑟, 𝑠) is the signature of the parent field (so 𝑛 = 𝑟+2𝑠
is the degree).
•prec (int) – desired floating point precision (default: default RealField precision).
•weighted (bool, default False) – if True, apply local degree weighting, i.e. double the value for
complex places.
OUTPUT:
(real) The archimedean local height of this number field element at the 𝑖‘th infinite place. If weighted
is True, this is multiplied by the local degree (as required for global heights), i.e. 1 for real places and 2
for complex places.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: [p.codomain() for p in K.places()]
[Real Field with 106 bits of precision,
Real Field with 106 bits of precision,
Complex Field with 53 bits of precision]
sage: [a.local_height_arch(i) for i in range(3)]
[0.5301924545717755083366563897519,
0.5301924545717755083366563897519,
0.886414217456333]
sage: [a.local_height_arch(i, weighted=True) for i in range(3)]
[0.5301924545717755083366563897519,
0.5301924545717755083366563897519,
1.77282843491267]
A relative example:
sage: L.<b, c> = NumberFieldTower([x^2 - 5, x^3 + x + 3])
sage: [(b + c).local_height_arch(i) for i in range(4)]
[1.238223390757884911842206617439,
0.02240347229957875780769746914391,
0.780028961749618,
1.16048938497298]
matrix ( base=None)
If base is None, return the matrix of right multiplication by the element on the power basis
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1, 𝑥, 𝑥2 , . . . , 𝑥𝑑−1 for the number field. Thus the rows of this matrix give the images of each of the
𝑥𝑖 .
If base is not None, then base must be either a field that embeds in the parent of self or a morphism to the
parent of self, in which case this function returns the matrix of multiplication by self on the power basis,
where we view the parent field as a field over base.
Specifying base as the base field over which the parent of self is a relative extension is equivalent to base
being None
INPUT:
•base - field or morphism
EXAMPLES:
Regular number field:
sage: K.<a> = NumberField(QQ['x'].0^3 - 5)
sage: M = a.matrix(); M
[0 1 0]
[0 0 1]
[5 0 0]
sage: M.base_ring() is QQ
True
Relative number field:
sage: L.<b> = K.extension(K['x'].0^2 - 2)
sage: M = b.matrix(); M
[0 1]
[2 0]
sage: M.base_ring() is K
True
Absolute number field:
sage: M = L.absolute_field('c').gen().matrix(); M
[ 0
1
0
0
0
0]
[ 0
0
1
0
0
0]
[ 0
0
0
1
0
0]
[ 0
0
0
0
1
0]
[ 0
0
0
0
0
1]
[-17 -60 -12 -10
6
0]
sage: M.base_ring() is QQ
True
More complicated relative number field:
sage: L.<b> = K.extension(K['x'].0^2 - a); L
Number Field in b with defining polynomial x^2 - a over its base field
sage: M = b.matrix(); M
[0 1]
[a 0]
sage: M.base_ring() is K
True
An example where we explicitly give the subfield or the embedding:
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sage: K.<a> = NumberField(x^4 + 1); L.<a2> = NumberField(x^2 + 1)
sage: a.matrix(L)
[ 0 1]
[a2 0]
Notice that if we compute all embeddings and choose a different one, then the matrix is changed as it
should be:
sage: v = L.embeddings(K)
sage: a.matrix(v[1])
[ 0
1]
[-a2
0]
The norm is also changed:
sage: a.norm(v[1])
a2
sage: a.norm(v[0])
-a2
minpoly ( var=’x’)
Return the minimal polynomial of this number field element.
EXAMPLES:
sage:
sage:
x^2 +
sage:
sage:
sage:
t^2 sage:
t^4 sage:
0
K.<a> = NumberField(x^2+3)
a.minpoly('x')
3
R.<X> = K['X']
L.<b> = K.extension(X^2-(22 + a))
b.minpoly('t')
a - 22
b.absolute_minpoly('t')
44*t^2 + 487
b^2 - (22+a)
multiplicative_order ( )
Return the multiplicative order of this number field element.
EXAMPLES:
sage: K.<z> = CyclotomicField(5)
sage: z.multiplicative_order()
5
sage: (-z).multiplicative_order()
10
sage: (1+z).multiplicative_order()
+Infinity
sage: x = polygen(QQ)
sage: K.<a>=NumberField(x^40 - x^20 + 4)
sage: u = 1/4*a^30 + 1/4*a^10 + 1/2
sage: u.multiplicative_order()
6
sage: a.multiplicative_order()
+Infinity
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An example in a relative extension:
sage:
sage:
sage:
12
sage:
True
K.<a, b> = NumberField([x^2 + x + 1, x^2 - 3])
z = (a - 1)*b/3
z.multiplicative_order()
z^12==1 and z^6!=1 and z^4!=1
norm ( K=None)
Return the absolute or relative norm of this number field element.
If K is given then K must be a subfield of the parent L of self, in which case the norm is the relative norm
from L to K. In all other cases, the norm is the absolute norm down to QQ.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + x^2 + x - 132/7); K
Number Field in a with defining polynomial x^3 + x^2 + x - 132/7
sage: a.norm()
132/7
sage: factor(a.norm())
2^2 * 3 * 7^-1 * 11
sage: K(0).norm()
0
Some complicated relatives norms in a tower of number fields.
sage:
sage:
sage:
1
sage:
1
sage:
1
sage:
a
sage:
121
sage:
2*c*b
sage:
-11
K.<a,b,c> = NumberField([x^2 + 1, x^2 + 3, x^2 + 5])
L = K.base_field(); M = L.base_field()
a.norm()
a.norm(L)
a.norm(M)
a
(a+b+c).norm()
(a+b+c).norm(L)
- 7
(a+b+c).norm(M)
We illustrate that norm is compatible with towers:
sage: z = (a+b+c).norm(L); z.norm(M)
-11
If we are in an order, the norm is an integer:
sage: K.<a> = NumberField(x^3-2)
sage: a.norm().parent()
Rational Field
sage: R = K.ring_of_integers()
sage: R(a).norm().parent()
Integer Ring
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When the base field is given by an embedding:
sage:
sage:
sage:
sage:
a2
sage:
-a2
K.<a> = NumberField(x^4 + 1)
L.<a2> = NumberField(x^2 + 1)
v = L.embeddings(K)
a.norm(v[1])
a.norm(v[0])
nth_root ( n, all=False)
Return an 𝑛‘th root of self in its parent 𝐾.
EXAMPLES:
sage: K.<a> = NumberField(x^4-7)
sage: K(7).nth_root(2)
a^2
sage: K((a-3)^5).nth_root(5)
a - 3
ALGORITHM: Use PARI to factor 𝑥𝑛 - self in 𝐾.
numerator_ideal ( )
Return the numerator ideal of this number field element.
The numerator ideal of a number field element 𝑎 is the ideal of the ring of integers 𝑅 obtained by intersecting 𝑎𝑅 with 𝑅.
See also:
denominator_ideal()
EXAMPLES:
sage: K.<a> = NumberField(x^2+5)
sage: b = (1+a)/2
sage: b.norm()
3/2
sage: N = b.numerator_ideal(); N
Fractional ideal (3, a + 1)
sage: N.norm()
3
sage: (1/b).numerator_ideal()
Fractional ideal (2, a + 1)
sage: K(0).numerator_ideal()
Ideal (0) of Number Field in a with defining polynomial x^2 + 5
ord ( P)
Returns the valuation of self at a given prime ideal P.
INPUT:
•P - a prime ideal of the parent of self
Note: The function ord() is an alias for valuation() .
EXAMPLES:
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sage:
sage:
sage:
sage:
sage:
1
sage:
1
sage:
<type
R.<x> = QQ[]
K.<a> = NumberField(x^4+3*x^2-17)
P = K.ideal(61).factor()[0][0]
b = a^2 + 30
b.valuation(P)
b.ord(P)
type(b.valuation(P))
'sage.rings.integer.Integer'>
The function can be applied to elements in relative number fields:
sage: L.<b> = K.extension(x^2 - 3)
sage: [L(6).valuation(P) for P in L.primes_above(2)]
[4]
sage: [L(6).valuation(P) for P in L.primes_above(3)]
[2, 2]
polynomial ( var=’x’)
Return the underlying polynomial corresponding to this number field element.
The resulting polynomial is currently not cached.
EXAMPLES:
sage: K.<a> = NumberField(x^5 - x - 1)
sage: f = (-2/3 + 1/3*a)^4; f
1/81*a^4 - 8/81*a^3 + 8/27*a^2 - 32/81*a + 16/81
sage: g = f.polynomial(); g
1/81*x^4 - 8/81*x^3 + 8/27*x^2 - 32/81*x + 16/81
sage: parent(g)
Univariate Polynomial Ring in x over Rational Field
Note that the result of this function is not cached (should this be changed?):
sage: g is f.polynomial()
False
relative_norm ( )
Return the relative norm of this number field element over the next field down in some tower of number
fields.
EXAMPLES:
sage: K1.<a1> = CyclotomicField(11)
sage: K2.<a2> = K1.extension(x^2 - 3)
sage: (a1 + a2).relative_norm()
a1^2 - 3
sage: (a1 + a2).relative_norm().relative_norm() == (a1 + a2).absolute_norm()
True
sage: K.<x,y,z> = NumberField([x^2 + 1, x^3 - 3, x^2 - 5])
sage: (x + y + z).relative_norm()
y^2 + 2*z*y + 6
residue_symbol ( P, m, check=True)
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The m-th power residue symbol for an element self and proper ideal P.
(︁ 𝛼 )︁
𝑁 (P)−1
≡ 𝛼 𝑚 mod P
P
Note: accepts m=1, in which case returns 1
Note: can also be called for an ideal from sage.rings.number_field_ideal.residue_symbol
Note: self is coerced into the number field of the ideal P
Note: if m=2, self is an integer, and P is an ideal of a number field of absolute degree 1 (i.e. it is a copy
of the rationals), then this calls kronecker_symbol, which is implemented using GMP.
INPUT:
•P - proper ideal of the number field (or an extension)
•m - positive integer
OUTPUT:
•an m-th root of unity in the number field
EXAMPLES:
Quadratic Residue (11 is not a square modulo 17):
sage: K.<a> = NumberField(x - 1)
sage: K(11).residue_symbol(K.ideal(17),2)
-1
sage: kronecker_symbol(11,17)
-1
The result depends on the number field of the ideal:
sage:
sage:
sage:
-1
sage:
1
K.<a> = NumberField(x - 1)
L.<b> = K.extension(x^2 + 1)
K(7).residue_symbol(K.ideal(11),2)
K(7).residue_symbol(L.ideal(11),2)
Cubic Residue:
sage: K.<w> = NumberField(x^2 - x + 1)
sage: (w^2 + 3).residue_symbol(K.ideal(17),3)
-w
The field must contain the m-th roots of unity:
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sage: K.<w> = NumberField(x^2 - x + 1)
sage: (w^2 + 3).residue_symbol(K.ideal(17),5)
Traceback (most recent call last):
...
ValueError: The residue symbol to that power is not defined for the number
˓→field
round ( )
Return the round (nearest integer) of this number field element.
EXAMPLES:
sage: x = polygen(ZZ)
sage: p = x**7 - 5*x**2 + x + 1
sage: a_AA = AA.polynomial_root(p, RIF(1,2))
sage: K.<a> = NumberField(p, embedding=a_AA)
sage: b = a**5 + a/2 - 1/7
sage: RR(b)
4.13444473767055
sage: b.round()
4
sage: (-b).round()
-4
sage: (b+1/2).round()
5
sage: (-b-1/2).round()
-5
This function always succeeds even if a tremendous precision is needed:
sage: c = b - 5678322907931/1225243417356 + 3
sage: c.round()
3
sage: RIF(c).unique_round()
Traceback (most recent call last):
...
ValueError: interval does not have a unique round (nearest integer)
If the number field is not embedded, this function is valid only if the element is rational:
sage: p = x**5 - 3
sage: K.<a> = NumberField(p)
sage: [K(k/3).round() for k in range(-3,4)]
[-1, -1, 0, 0, 0, 1, 1]
sage: a.round()
Traceback (most recent call last):
...
TypeError: floor not uniquely defined since no real embedding is specified
sign ( )
Return the sign of this algebraic number (if a real embedding is well defined)
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 2, embedding=AA(2)**(1/3))
sage: K.zero().sign()
0
sage: K.one().sign()
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sage:
-1
sage:
1
sage:
1
sage:
-1
(-K.one()).sign()
a.sign()
(a - 234917380309015/186454048314072).sign()
(a - 3741049304830488/2969272800976409).sign()
If the field is not embedded in real numbers, this method will only work for rational elements:
sage: L.<b> = NumberField(x^4 - x - 1)
sage: b.sign()
Traceback (most recent call last):
...
TypeError: sign not well defined since no real embedding is
specified
sage: L(-33/125).sign()
-1
sage: L.zero().sign()
0
sqrt ( all=False)
Returns the square root of this number in the given number field.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 3)
sage: K(3).sqrt()
a
sage: K(3).sqrt(all=True)
[a, -a]
sage: K(a^10).sqrt()
9*a
sage: K(49).sqrt()
7
sage: K(1+a).sqrt()
Traceback (most recent call last):
...
ValueError: a + 1 not a square in Number Field in a with defining polynomial
˓→x^2 - 3
sage: K(0).sqrt()
0
sage: K((7+a)^2).sqrt(all=True)
[a + 7, -a - 7]
sage: K.<a> = CyclotomicField(7)
sage: a.sqrt()
a^4
sage: K.<a> = NumberField(x^5 - x + 1)
sage: (a^4 + a^2 - 3*a + 2).sqrt()
a^3 - a^2
ALGORITHM: Use PARI to factor 𝑥2 - self in 𝐾.
support ( )
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Return the support of this number field element.
OUTPUT: A sorted list of the primes ideals at which this number field element has nonzero valuation. An
error is raised if the element is zero.
EXAMPLES:
sage: x = ZZ['x'].gen()
sage: F.<t> = NumberField(x^3 - 2)
sage: P5s = F(5).support()
sage: P5s
[Fractional ideal (-t^2 - 1), Fractional ideal (t^2 - 2*t - 1)]
sage: all(5 in P5 for P5 in P5s)
True
sage: all(P5.is_prime() for P5 in P5s)
True
sage: [ P5.norm() for P5 in P5s ]
[5, 25]
trace ( K=None)
Return the absolute or relative trace of this number field element.
If K is given then K must be a subfield of the parent L of self, in which case the trace is the relative trace
from L to K. In all other cases, the trace is the absolute trace down to QQ.
EXAMPLES:
sage: K.<a> = NumberField(x^3 -132/7*x^2 + x + 1); K
Number Field in a with defining polynomial x^3 - 132/7*x^2 + x + 1
sage: a.trace()
132/7
sage: (a+1).trace() == a.trace() + 3
True
If we are in an order, the trace is an integer:
sage: K.<zeta> = CyclotomicField(17)
sage: R = K.ring_of_integers()
sage: R(zeta).trace().parent()
Integer Ring
valuation ( P)
Returns the valuation of self at a given prime ideal P.
INPUT:
•P - a prime ideal of the parent of self
Note: The function ord() is an alias for valuation() .
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
R.<x> = QQ[]
K.<a> = NumberField(x^4+3*x^2-17)
P = K.ideal(61).factor()[0][0]
b = a^2 + 30
b.valuation(P)
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sage: b.ord(P)
1
sage: type(b.valuation(P))
<type 'sage.rings.integer.Integer'>
The function can be applied to elements in relative number fields:
sage: L.<b> = K.extension(x^2 - 3)
sage: [L(6).valuation(P) for P in L.primes_above(2)]
[4]
sage: [L(6).valuation(P) for P in L.primes_above(3)]
[2, 2]
vector ( )
Return vector representation of self in terms of the basis for the ambient number field.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 1)
sage: (2/3*a - 5/6).vector()
(-5/6, 2/3)
sage: (-5/6, 2/3)
(-5/6, 2/3)
sage: O = K.order(2*a)
sage: (O.1).vector()
(0, 2)
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: (a + b).vector()
(b, 1)
sage: O = K.order([a,b])
sage: (O.1).vector()
(-b, 1)
sage: (O.2).vector()
(1, -b)
class sage.rings.number_field.number_field_element. NumberFieldElement_absolute
Bases: sage.rings.number_field.number_field_element.NumberFieldElement
absolute_charpoly ( var=’x’, algorithm=None)
Return the characteristic polynomial of this element over Q.
For the meaning of the optional argument algorithm , see charpoly() .
EXAMPLES:
sage:
sage:
sage:
x^4 +
sage:
y^4 +
sage:
x^4 +
sage:
x^2 +
x = ZZ['x'].0
K.<a> = NumberField(x^4 + 2, 'a')
a.absolute_charpoly()
2
a.absolute_charpoly('y')
2
(-a^2).absolute_charpoly()
4*x^2 + 4
(-a^2).absolute_minpoly()
2
sage: a.absolute_charpoly(algorithm='pari') == a.absolute_charpoly(algorithm=
˓→'sage')
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True
absolute_minpoly ( var=’x’, algorithm=None)
Return the minimal polynomial of this element over Q.
For the meaning of the optional argument algorithm, see charpoly() .
EXAMPLES:
sage: x = ZZ['x'].0
sage: f = x^10 - 5*x^9 + 15*x^8 - 68*x^7 + 81*x^6 - 221*x^5 + 141*x^4 - 242*x^
˓→3 - 13*x^2 - 33*x - 135
sage: K.<a> = NumberField(f, 'a')
sage: a.absolute_charpoly()
x^10 - 5*x^9 + 15*x^8 - 68*x^7 + 81*x^6 - 221*x^5 + 141*x^4 - 242*x^3 - 13*x^
˓→2 - 33*x - 135
sage: a.absolute_charpoly('y')
y^10 - 5*y^9 + 15*y^8 - 68*y^7 + 81*y^6 - 221*y^5 + 141*y^4 - 242*y^3 - 13*y^
˓→2 - 33*y - 135
sage: b = -79/9995*a^9 + 52/9995*a^8 + 271/9995*a^7 + 1663/9995*a^6 + 13204/
˓→9995*a^5 + 5573/9995*a^4 + 8435/1999*a^3 - 3116/9995*a^2 + 7734/1999*a +
˓→1620/1999
sage: b.absolute_charpoly()
x^10 + 10*x^9 + 25*x^8 - 80*x^7 - 438*x^6 + 80*x^5 + 2950*x^4 + 1520*x^3 ˓→10439*x^2 - 5130*x + 18225
sage: b.absolute_minpoly()
x^5 + 5*x^4 - 40*x^2 - 19*x + 135
sage: b.absolute_minpoly(algorithm='pari') == b.absolute_minpoly(algorithm=
˓→'sage')
True
charpoly ( var=’x’, algorithm=None)
The characteristic polynomial of this element, over Q if self is an element of a field, and over Z is self is
an element of an order.
This is the same as self.absolute_charpoly since this is an element of an absolute extension.
The optional argument algorithm controls how the characteristic polynomial is computed: ‘pari’ uses
PARI, ‘sage’ uses charpoly for Sage matrices. The default value None means that ‘pari’ is used for small
degrees (up to the value of the constant TUNE_CHARPOLY_NF, currently at 25), otherwise ‘sage’ is used.
The constant TUNE_CHARPOLY_NF should give reasonable performance on all architectures; however,
if you feel the need to customize it to your own machine, see trac ticket #5213 for a tuning script.
EXAMPLES:
We compute the characteristic polynomial of the cube root of 2.
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3-2)
sage: a.charpoly('x')
x^3 - 2
sage: a.charpoly('y').parent()
Univariate Polynomial Ring in y over Rational Field
is_real_positive ( min_prec=53)
Using the n method of approximation, return True if self is a real positive number and False
otherwise. This method is completely dependent of the embedding used by the n method.
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The algorithm first checks that self is not a strictly complex number. Then if self is not zero,
by approximation more and more precise, the method answers True if the number is positive. Using
𝑅𝑒𝑎𝑙𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙, the result is guaranteed to be correct.
For CyclotomicField, the embedding is the natural one sending 𝑧𝑒𝑡𝑎𝑛 on 𝑐𝑜𝑠(2 * 𝑝𝑖/𝑛).
EXAMPLES:
sage:
sage:
False
sage:
True
sage:
sage:
True
sage:
sage:
sage:
True
sage:
sage:
False
sage:
sage:
True
sage:
sage:
False
sage:
False
K.<a> = CyclotomicField(3)
(a+a^2).is_real_positive()
(-a-a^2).is_real_positive()
K.<a> = CyclotomicField(1000)
(a+a^(-1)).is_real_positive()
K.<a> = CyclotomicField(1009)
d = a^252
(d+d.conjugate()).is_real_positive()
d = a^253
(d+d.conjugate()).is_real_positive()
K.<a> = QuadraticField(3)
a.is_real_positive()
K.<a> = QuadraticField(-3)
a.is_real_positive()
(a-a).is_real_positive()
lift ( var=’x’)
Return an element of QQ[x], where this number field element lives in QQ[x]/(f(x)).
EXAMPLES:
sage: K.<a> = QuadraticField(-3)
sage: a.lift()
x
list ( )
Return the list of coefficients of self written in terms of a power basis.
EXAMPLES:
sage: K.<z> = CyclotomicField(3)
sage: (2+3/5*z).list()
[2, 3/5]
sage: (5*z).list()
[0, 5]
sage: K(3).list()
[3, 0]
minpoly ( var=’x’, algorithm=None)
Return the minimal polynomial of this number field element.
For the meaning of the optional argument algorithm, see charpoly().
EXAMPLES:
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We compute the characteristic polynomial of cube root of 2.
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3-2)
sage: a.minpoly('x')
x^3 - 2
sage: a.minpoly('y').parent()
Univariate Polynomial Ring in y over Rational Field
class sage.rings.number_field.number_field_element. NumberFieldElement_relative
Bases: sage.rings.number_field.number_field_element.NumberFieldElement
The current relative number field element implementation does everything in terms of absolute polynomials.
All conversions from relative polynomials, lists, vectors, etc should happen in the parent.
absolute_charpoly ( var=’x’, algorithm=None)
The characteristic polynomial of this element over Q.
We construct a relative extension and find the characteristic polynomial over Q.
The optional argument algorithm controls how the characteristic polynomial is computed: ‘pari’ uses
PARI, ‘sage’ uses charpoly for Sage matrices. The default value None means that ‘pari’ is used for small
degrees (up to the value of the constant TUNE_CHARPOLY_NF, currently at 25), otherwise ‘sage’ is used.
The constant TUNE_CHARPOLY_NF should give reasonable performance on all architectures; however,
if you feel the need to customize it to your own machine, see trac ticket #5213 for a tuning script.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^3-2)
sage: S.<X> = K[]
sage: L.<b> = NumberField(X^3 + 17); L
Number Field in b with defining polynomial X^3 + 17 over its base field
sage: b.absolute_charpoly()
x^9 + 51*x^6 + 867*x^3 + 4913
sage: b.charpoly()(b)
0
sage: a = L.0; a
b
sage: a.absolute_charpoly('x')
x^9 + 51*x^6 + 867*x^3 + 4913
sage: a.absolute_charpoly('y')
y^9 + 51*y^6 + 867*y^3 + 4913
sage: a.absolute_charpoly(algorithm='pari') == a.absolute_charpoly(algorithm=
˓→'sage')
True
absolute_minpoly ( var=’x’, algorithm=None)
Return the minimal polynomial over Q of this element.
For the meaning of the optional argument algorithm , see absolute_charpoly() .
EXAMPLES:
sage:
sage:
sage:
sage:
x^8 -
K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
y = K['y'].0
L.<c> = K.extension(y^2 + a*y + b)
c.absolute_charpoly()
1996*x^6 + 996006*x^4 + 1997996*x^2 + 1
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sage: c.absolute_minpoly()
x^8 - 1996*x^6 + 996006*x^4 + 1997996*x^2 + 1
sage: L(a).absolute_charpoly()
x^8 + 8*x^6 + 24*x^4 + 32*x^2 + 16
sage: L(a).absolute_minpoly()
x^2 + 2
sage: L(b).absolute_charpoly()
x^8 + 4000*x^7 + 6000004*x^6 + 4000012000*x^5 + 1000012000006*x^4 +
˓→4000012000*x^3 + 6000004*x^2 + 4000*x + 1
sage: L(b).absolute_minpoly()
x^2 + 1000*x + 1
charpoly ( var=’x’)
The characteristic polynomial of this element over its base field.
EXAMPLES:
sage:
sage:
sage:
x^2 +
sage:
x^2 sage:
x - b
x = ZZ['x'].0
K.<a, b> = QQ.extension([x^2 + 2, x^5 + 400*x^4 + 11*x^2 + 2])
a.charpoly()
2
b.charpoly()
2*b*x + b^2
b.minpoly()
sage:
sage:
sage:
sage:
x^2 +
sage:
x^2 +
sage:
x^2 sage:
x - a
sage:
x^2 sage:
x - b
K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
y = K['y'].0
L.<c> = K.extension(y^2 + a*y + b)
c.charpoly()
a* x + b
c.minpoly()
a* x + b
L(a).charpoly()
2*a*x - 2
L(a).minpoly()
L(b).charpoly()
2*b*x - 1000*b - 1
L(b).minpoly()
lift ( var=’x’)
Return an element of K[x], where this number field element lives in the relative number field K[x]/(f(x)).
EXAMPLES:
sage: K.<a> = QuadraticField(-3)
sage: x = polygen(K)
sage: L.<b> = K.extension(x^7 + 5)
sage: u = L(1/2*a + 1/2 + b + (a-9)*b^5)
sage: u.lift()
(a - 9)*x^5 + x + 1/2*a + 1/2
list ( )
Return the list of coefficients of self written in terms of a power basis.
EXAMPLES:
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sage: K.<a,b> = NumberField([x^3+2, x^2+1])
sage: a.list()
[0, 1, 0]
sage: v = (K.base_field().0 + a)^2 ; v
a^2 + 2*b*a - 1
sage: v.list()
[-1, 2*b, 1]
valuation ( P)
Returns the valuation of self at a given prime ideal P.
INPUT:
•P - a prime ideal of relative number field which is the parent of self
EXAMPLES:
sage: K.<a, b, c> = NumberField([x^2 - 2, x^2 - 3, x^2 - 5])
sage: P = K.prime_factors(5)[0]
sage: (2*a + b - c).valuation(P)
1
class sage.rings.number_field.number_field_element. OrderElement_absolute
Bases: sage.rings.number_field.number_field_element.NumberFieldElement_absolute
Element of an order in an absolute number field.
EXAMPLES:
sage:
sage:
sage:
2*a
sage:
Order
K.<a> = NumberField(x^2 + 1)
O2 = K.order(2*a)
w = O2.1; w
parent(w)
in Number Field in a with defining polynomial x^2 + 1
sage: w.absolute_charpoly()
x^2 + 4
sage: w.absolute_charpoly().parent()
Univariate Polynomial Ring in x over Integer Ring
sage: w.absolute_minpoly()
x^2 + 4
sage: w.absolute_minpoly().parent()
Univariate Polynomial Ring in x over Integer Ring
inverse_mod ( I)
Return an inverse of self modulo the given ideal.
INPUT:
•I - may be an ideal of self.parent(), or an element or list of elements of self.parent() generating a
nonzero ideal. A ValueError is raised if I is non-integral or is zero. A ZeroDivisionError is raised if I
+ (x) != (1).
EXAMPLES:
sage: OE.<w> = EquationOrder(x^3 - x + 2)
sage: w.inverse_mod(13*OE)
6*w^2 - 6
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sage: w * (w.inverse_mod(13)) - 1 in 13*OE
True
sage: w.inverse_mod(13).parent() == OE
True
sage: w.inverse_mod(2*OE)
Traceback (most recent call last):
...
ZeroDivisionError: w is not invertible modulo Fractional ideal (2)
class sage.rings.number_field.number_field_element. OrderElement_relative
Bases: sage.rings.number_field.number_field_element.NumberFieldElement_relative
Element of an order in a relative number field.
EXAMPLES:
sage: O = EquationOrder([x^2 + x + 1, x^3 - 2],'a,b')
sage: c = O.1; c
(-2*b^2 - 2)*a - 2*b^2 - b
sage: type(c)
<type 'sage.rings.number_field.number_field_element.OrderElement_relative'>
absolute_charpoly ( var=’x’)
The absolute characteristic polynomial of this order element over ZZ.
EXAMPLES:
sage: x = ZZ['x'].0
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order()
sage: _, u, _, v = OK.basis()
sage: t = 2*u - v; t
-b
sage: t.absolute_charpoly()
x^4 - 6*x^2 + 9
sage: t.absolute_minpoly()
x^2 - 3
sage: t.absolute_charpoly().parent()
Univariate Polynomial Ring in x over Integer Ring
absolute_minpoly ( var=’x’)
The absolute minimal polynomial of this order element over ZZ.
EXAMPLES:
sage: x = ZZ['x'].0
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order()
sage: _, u, _, v = OK.basis()
sage: t = 2*u - v; t
-b
sage: t.absolute_charpoly()
x^4 - 6*x^2 + 9
sage: t.absolute_minpoly()
x^2 - 3
sage: t.absolute_minpoly().parent()
Univariate Polynomial Ring in x over Integer Ring
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charpoly ( var=’x’)
The characteristic polynomial of this order element over its base ring.
This special implementation works around bug #4738. At this time the base ring of relative order elements
is ZZ; it should be the ring of integers of the base field.
EXAMPLES:
sage: x = ZZ['x'].0
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order(); OK.basis()
[1, 1/2*a - 1/2*b, -1/2*b*a + 1/2, a]
sage: charpoly(OK.1)
x^2 + b*x + 1
sage: charpoly(OK.1).parent()
Univariate Polynomial Ring in x over Maximal Order in Number Field in b with
˓→defining polynomial x^2 - 3
sage: [ charpoly(t) for t in OK.basis() ]
[x^2 - 2*x + 1, x^2 + b*x + 1, x^2 - x + 1, x^2 + 1]
inverse_mod ( I)
Return an inverse of self modulo the given ideal.
INPUT:
•I - may be an ideal of self.parent(), or an element or list of elements of self.parent() generating a
nonzero ideal. A ValueError is raised if I is non-integral or is zero. A ZeroDivisionError is raised if I
+ (x) != (1).
EXAMPLES:
sage:
sage:
sage:
sage:
True
sage:
True
E.<a,b> = NumberField([x^2 - x + 2, x^2 + 1])
OE = E.ring_of_integers()
t = OE(b - a).inverse_mod(17*b)
t*(b - a) - 1 in E.ideal(17*b)
t.parent() == OE
minpoly ( var=’x’)
The minimal polynomial of this order element over its base ring.
This special implementation works around bug #4738. At this time the base ring of relative order elements
is ZZ; it should be the ring of integers of the base field.
EXAMPLES:
sage: x = ZZ['x'].0
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
sage: OK = K.maximal_order(); OK.basis()
[1, 1/2*a - 1/2*b, -1/2*b*a + 1/2, a]
sage: minpoly(OK.1)
x^2 + b*x + 1
sage: charpoly(OK.1).parent()
Univariate Polynomial Ring in x over Maximal Order in Number Field in b with
˓→defining polynomial x^2 - 3
sage: _, u, _, v = OK.basis()
sage: t = 2*u - v; t
-b
sage: t.charpoly()
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x^2 + 2*b*x + 3
sage: t.minpoly()
x + b
sage:
x^4 sage:
x^2 -
t.absolute_charpoly()
6*x^2 + 9
t.absolute_minpoly()
3
sage.rings.number_field.number_field_element. is_NumberFieldElement ( x)
Return True if x is of type NumberFieldElement, i.e., an element of a number field.
EXAMPLES:
sage: from sage.rings.number_field.number_field_element import is_
˓→NumberFieldElement
sage: is_NumberFieldElement(2)
False
sage: k.<a> = NumberField(x^7 + 17*x + 1)
sage: is_NumberFieldElement(a+1)
True
1.5 Optimized Quadratic Number Field Elements
This file defines a Cython class NumberFieldElement_quadratic to speed up computations in quadratic
extensions of Q.
AUTHORS:
• Robert Bradshaw (2007-09): Initial version
• David Harvey (2007-10): fix up a few bugs, polish around the edges
• David Loeffler (2009-05): add more documentation and tests
• Vincent Delecroix (2012-07): comparisons for quadratic number fields (trac ticket #13213), abs, floor and ceil
functions (trac ticket #13256)
Todo
The _new() method should be overridden in this class to copy the D and standard_embedding attributes
class sage.rings.number_field.number_field_element_quadratic. NumberFieldElement_quadratic
Bases: sage.rings.number_field.number_field_element.NumberFieldElement_absolute
A NumberFieldElement_quadratic object gives an efficient representation of an element of a quadratic extension
of Q.
Elements are represented
internally as triples (𝑎, 𝑏, 𝑐) of integers, where gcd(𝑎, 𝑏, 𝑐) = 1 and 𝑐 > 0, representing
√
the element (𝑎 + 𝑏 𝐷)/𝑐. Note that if the discriminant 𝐷 is 1 mod 4, integral elements do not necessarily have
𝑐 = 1.
ceil ( )
Returns the ceil.
EXAMPLES:
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sage:
sage:
3
sage:
-2
sage:
9
K.<sqrt7> = QuadraticField(7, name='sqrt7')
sqrt7.ceil()
(-sqrt7).ceil()
(1022/313*sqrt7 - 14/23).ceil()
charpoly ( var=’x’)
The characteristic polynomial of this element over Q.
EXAMPLES:
sage:
sage:
x^2 sage:
sage:
x^2 sage:
0
K.<a> = NumberField(x^2-x+13)
a.charpoly()
x + 13
b = 3-a/2
f = b.charpoly(); f
11/2*x + 43/4
f(b)
continued_fraction ( )
Return the (finite or ultimately periodic) continued fraction of self .
EXAMPLES:
sage: K.<sqrt2> = QuadraticField(2)
sage: cf = sqrt2.continued_fraction(); cf
[1; (2)*]
sage: cf.n()
1.41421356237310
sage: sqrt2.n()
1.41421356237309
sage: cf.value()
sqrt2
sage: (sqrt2/3 + 1/4).continued_fraction()
[0; 1, (2, 1, 1, 2, 3, 2, 1, 1, 2, 5, 1, 1, 14, 1, 1, 5)*]
continued_fraction_list ( )
Return the preperiod and the period of the continued fraction expansion of self .
EXAMPLES:
sage: K.<sqrt2> = QuadraticField(2)
sage: sqrt2.continued_fraction_list()
((1,), (2,))
sage: (1/2+sqrt2/3).continued_fraction_list()
((0, 1, 33), (1, 32))
For rational entries a pair of tuples is also returned but the second one is empty:
sage: K(123/567).continued_fraction_list()
((0, 4, 1, 1, 1, 1, 3, 2), ())
denominator ( )
Return the denominator of self. This is the LCM of the denominators of the coefficients of self, and thus it
may well be > 1 even when the element is an algebraic integer.
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EXAMPLES:
sage:
sage:
1
sage:
sage:
6
sage:
1
sage:
2
sage:
1
K.<a> = NumberField(x^2+x+41)
a.denominator()
sage:
sage:
sage:
2
sage:
True
K.<a> = NumberField(x^2 - 5)
b = (a + 1)/2
b.denominator()
b = (2*a+1)/6
b.denominator()
K(1).denominator()
K(1/2).denominator()
K(0).denominator()
b.is_integral()
floor ( )
Returns the floor of x.
EXAMPLES:
sage:
sage:
1
sage:
-2
sage:
42
sage:
-43
K.<sqrt2> = QuadraticField(2,name='sqrt2')
sqrt2.floor()
(-sqrt2).floor()
(13/197 + 3702/123*sqrt2).floor()
(13/197-3702/123*sqrt2).floor()
galois_conjugate ( )
Return the image of this element under action of the nontrivial element of the Galois group of this field.
EXAMPLES:
sage: K.<a> = QuadraticField(23)
sage: a.galois_conjugate()
-a
sage: K.<a> = NumberField(x^2 - 5*x + 1)
sage: a.galois_conjugate()
-a + 5
sage: b = 5*a + 1/3
sage: b.galois_conjugate()
-5*a + 76/3
sage: b.norm() == b * b.galois_conjugate()
True
sage: b.trace() == b + b.galois_conjugate()
True
imag ( )
Returns the imaginary part of self.
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EXAMPLES:
sage: K.<sqrt2> = QuadraticField(2)
sage: sqrt2.imag()
0
sage: parent(sqrt2.imag())
Rational Field
sage: K.<i> = QuadraticField(-1)
sage: i.imag()
1
sage: parent(i.imag())
Rational Field
sage: K.<a> = NumberField(x^2 + x + 1, embedding=CDF.0)
sage: a.imag()
1/2*sqrt3
sage: a.real()
-1/2
sage: SR(a)
1/2*I*sqrt(3) - 1/2
sage: bool(I*a.imag() + a.real() == a)
True
is_integer ( )
Check whether this number field element is an integer.
See also:
•is_rational() to test if this element is a rational number
•is_integral() to test if this element is an algebraic integer
EXAMPLES:
sage:
sage:
False
sage:
False
sage:
True
sage:
True
sage:
False
K.<sqrt3> = QuadraticField(3)
sqrt3.is_integer()
(sqrt3-1/2).is_integer()
K(0).is_integer()
K(-12).is_integer()
K(1/3).is_integer()
is_integral ( )
Returns whether this element is an algebraic integer.
is_one ( )
Check whether this number field element is 1.
EXAMPLES:
sage: K = QuadraticField(-2)
sage: K(1).is_one()
True
sage: K(-1).is_one()
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False
sage:
False
sage:
False
sage:
False
sage:
False
K(2).is_one()
K(0).is_one()
K(1/2).is_one()
K.gen().is_one()
is_rational ( )
Check whether this number field element is a rational number.
See also:
•is_integer() to test if this element is an integer
•is_integral() to test if this element is an algebraic integer
EXAMPLES:
sage:
sage:
False
sage:
False
sage:
True
sage:
True
sage:
True
K.<sqrt3> = QuadraticField(3)
sqrt3.is_rational()
(sqrt3-1/2).is_rational()
K(0).is_rational()
K(-12).is_rational()
K(1/3).is_rational()
minpoly ( var=’x’)
The minimal polynomial of this element over Q.
INPUT:
•var – the minimal polynomial is defined over a polynomial ring in a variable with this name. If not
specified this defaults to x .
EXAMPLES:
sage: K.<a> = NumberField(x^2+13)
sage: a.minpoly()
x^2 + 13
sage: a.minpoly('T')
T^2 + 13
sage: (a+1/2-a).minpoly()
x - 1/2
norm ( K=None)
Return the norm of self. If the second argument is None, this is the norm down to Q. Otherwise, return
the norm down to K (which had better be either Q or this number field).
EXAMPLES:
sage: K.<a> = NumberField(x^2-x+3)
sage: a.norm()
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sage: a.matrix()
[ 0 1]
[-3 1]
sage: K.<a> = NumberField(x^2+5)
sage: (1+a).norm()
6
The norm is multiplicative:
sage:
sage:
-3
sage:
9
sage:
-27
K.<a> = NumberField(x^2-3)
a.norm()
K(3).norm()
(3*a).norm()
We test that the optional argument is handled sensibly:
sage: (3*a).norm(QQ)
-27
sage: (3*a).norm(K)
3*a
sage: (3*a).norm(CyclotomicField(3))
Traceback (most recent call last):
...
ValueError: no way to embed L into parent's base ring K
numerator ( )
Return self*self.denominator().
EXAMPLES:
sage:
sage:
sage:
6
sage:
2*a +
K.<a> = NumberField(x^2+x+41)
b = (2*a+1)/6
b.denominator()
b.numerator()
1
parts ( )
√
This function returns a pair of rationals 𝑎 and 𝑏 such that self = 𝑎 + 𝑏 𝐷.
This is much closer to the internal storage format of the elements than the polynomial representation coefficients (the output
of self.list() ), unless the generator with which this number field was constructed
√
was equal to 𝐷. See the last example below.
EXAMPLES:
sage: K.<a> = NumberField(x^2-13)
sage: K.discriminant()
13
sage: a.parts()
(0, 1)
sage: (a/2-4).parts()
(-4, 1/2)
sage: K.<a> = NumberField(x^2-7)
sage: K.discriminant()
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sage: a.parts()
(0, 1)
sage: K.<a> = NumberField(x^2-x+7)
sage: a.parts()
(1/2, 3/2)
sage: a._coefficients()
[0, 1]
real ( )
Returns the real part of self, which is either self (if self lives it a totally real field) or a rational number.
EXAMPLES:
sage: K.<sqrt2> = QuadraticField(2)
sage: sqrt2.real()
sqrt2
sage: K.<a> = QuadraticField(-3)
sage: a.real()
0
sage: (a + 1/2).real()
1/2
sage: K.<a> = NumberField(x^2 + x + 1)
sage: a.real()
-1/2
sage: parent(a.real())
Rational Field
sage: K.<i> = QuadraticField(-1)
sage: i.real()
0
round ( )
Returns the round (nearest integer).
EXAMPLES:
sage:
sage:
3
sage:
-3
sage:
0
sage:
-1
K.<sqrt7> = QuadraticField(7, name='sqrt7')
sqrt7.round()
(-sqrt7).round()
(12/313*sqrt7 - 1745917/2902921).round()
(12/313*sqrt7 - 1745918/2902921).round()
sign ( )
Returns the sign of self (0 if zero, +1 if positive and -1 if negative).
EXAMPLES:
sage:
sage:
0
sage:
1
sage:
1
sage:
158
K.<sqrt2> = QuadraticField(2, name='sqrt2')
K(0).sign()
sqrt2.sign()
(sqrt2+1).sign()
(sqrt2-1).sign()
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sage:
-1
sage:
-1
sage:
-1
sage:
1
sage:
sage:
0
sage:
-1
sage:
-1
sage:
1
sage:
-1
sage:
1
sage:
1
sage:
-1
(sqrt2-2).sign()
(-sqrt2).sign()
(-sqrt2+1).sign()
(-sqrt2+2).sign()
K.<a> = QuadraticField(2, embedding=-1.4142)
K(0).sign()
a.sign()
(a+1).sign()
(a+2).sign()
(a-1).sign()
(-a).sign()
(-a-1).sign()
(-a-2).sign()
sage: K.<b> = NumberField(x^2 + 2*x + 7, 'b', embedding=CC(-1,-sqrt(6)))
sage: b.sign()
Traceback (most recent call last):
...
ValueError: a complex number has no sign!
sage: K(1).sign()
1
sage: K(0).sign()
0
sage: K(-2/3).sign()
-1
trace ( )
EXAMPLES:
sage:
sage:
-1
sage:
[ 0
[-41
K.<a> = NumberField(x^2+x+41)
a.trace()
a.matrix()
1]
-1]
The trace is additive:
sage:
sage:
2
sage:
6
sage:
K.<a> = NumberField(x^2+7)
(a+1).trace()
K(3).trace()
(a+4).trace()
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sage: (a/3+1).trace()
2
class sage.rings.number_field.number_field_element_quadratic. OrderElement_quadratic
Bases: sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadrati
Element of an order in a quadratic field.
EXAMPLES:
sage:
sage:
sage:
2*a
sage:
Order
K.<a> = NumberField(x^2 + 1)
O2 = K.order(2*a)
w = O2.1; w
parent(w)
in Number Field in a with defining polynomial x^2 + 1
charpoly ( var=’x’)
The characteristic polynomial of this element, which is over Z because this element is an algebraic integer.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 5)
sage: R = K.ring_of_integers()
sage: b = R((5+a)/2)
sage: f = b.charpoly('x'); f
x^2 - 5*x + 5
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: f(b)
0
inverse_mod ( I)
Return an inverse of self modulo the given ideal.
INPUT:
•I - may be an ideal of self.parent(), or an element or list of elements of self.parent() generating a
nonzero ideal. A ValueError is raised if I is non-integral or is zero. A ZeroDivisionError is raised if I
+ (x) != (1).
EXAMPLES:
sage: OE.<w> = EquationOrder(x^2 - x + 2)
sage: w.inverse_mod(13) == 6*w - 6
True
sage: w*(6*w - 6) - 1
-13
sage: w.inverse_mod(13).parent() == OE
True
sage: w.inverse_mod(2*OE)
Traceback (most recent call last):
...
ZeroDivisionError: w is not invertible modulo Fractional ideal (2)
minpoly ( var=’x’)
The minimal polynomial of this element over Z.
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EXAMPLES:
sage: K.<a> = NumberField(x^2 + 163)
sage: R = K.ring_of_integers()
sage: f = R(a).minpoly('x'); f
x^2 + 163
sage: f.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: R(5).minpoly()
x - 5
norm ( )
The norm of an element of the ring of integers is an Integer.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 3)
sage: O2 = K.order(2*a)
sage: w = O2.gen(1); w
2*a
sage: w.norm()
12
sage: parent(w.norm())
Integer Ring
trace ( )
The trace of an element of the ring of integers is an Integer.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 5)
sage: R = K.ring_of_integers()
sage: b = R((1+a)/2)
sage: b.trace()
1
sage: parent(b.trace())
Integer Ring
class sage.rings.number_field.number_field_element_quadratic. Q_to_quadratic_field_element
Bases: sage.categories.morphism.Morphism
Morphism that coerces from rationals to elements of a quadratic number field K.
class sage.rings.number_field.number_field_element_quadratic. Z_to_quadratic_field_element
Bases: sage.categories.morphism.Morphism
Morphism that coerces from integers to elements of a quadratic number field K.
1.6 Splitting fields of polynomials over number fields
AUTHORS:
• Jeroen Demeyer (2014-01-02): initial version for trac ticket #2217
• Jeroen Demeyer (2014-01-03): add abort_degree argument, trac ticket #15626
class sage.rings.number_field.splitting_field. SplittingData ( _pol, _dm)
A class to store data for internal use in splitting_field() . It contains two attributes pol (polynomial),
dm (degree multiple), where pol is a PARI polynomial and dm a Sage Integer .
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dm is a multiple of the degree of the splitting field of pol over some field 𝐸. In splitting_field() , 𝐸
is the field containing the current field 𝐾 and all roots of other polynomials inside the list 𝐿 with dm less than
this dm .
key ( )
Return a sorting key. Compare first by degree bound, then by polynomial degree, then by discriminant.
EXAMPLES:
sage: from sage.rings.number_field.splitting_field import SplittingData
sage: L = []
sage: L.append(SplittingData(pari("x^2 + 1"), 1))
sage: L.append(SplittingData(pari("x^3 + 1"), 1))
sage: L.append(SplittingData(pari("x^2 + 7"), 2))
sage: L.append(SplittingData(pari("x^3 + 1"), 2))
sage: L.append(SplittingData(pari("x^3 + x^2 + x + 1"), 2))
sage: L.sort(key=lambda x: x.key()); L
[SplittingData(x^2 + 1, 1), SplittingData(x^3 + 1, 1), SplittingData(x^2 + 7,
˓→2), SplittingData(x^3 + x^2 + x + 1, 2), SplittingData(x^3 + 1, 2)]
sage: [x.key() for x in L]
[(1, 2, 16), (1, 3, 729), (2, 2, 784), (2, 3, 256), (2, 3, 729)]
poldegree ( )
Return the degree of self.pol
EXAMPLES:
sage: from sage.rings.number_field.splitting_field import SplittingData
sage: SplittingData(pari("x^123 + x + 1"), 2).poldegree()
123
exception sage.rings.number_field.splitting_field. SplittingFieldAbort ( div,
mult)
Bases: exceptions.Exception
Special exception class to indicate an early abort of splitting_field() .
EXAMPLES:
sage: from sage.rings.number_field.splitting_field import SplittingFieldAbort
sage: raise SplittingFieldAbort(20, 60)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field is a multiple of 20
sage: raise SplittingFieldAbort(12, 12)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field equals 12
sage.rings.number_field.splitting_field. splitting_field ( poly, name, map=False,
degree_multiple=None,
abort_degree=None,
simplify=True,
simplify_all=False)
Compute the splitting field of a given polynomial, defined over a number field.
INPUT:
•poly – a monic polynomial over a number field
•name – a variable name for the number field
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•map – (default: False ) also return an embedding of poly into the resulting field. Note that computing
this embedding might be expensive.
•degree_multiple – a multiple of the absolute degree of the splitting field. If degree_multiple
equals the actual degree, this can enormously speed up the computation.
•abort_degree – abort by raising a SplittingFieldAbort if it can be determined that the absolute degree of the splitting field is strictly larger than abort_degree .
•simplify – (default: True ) during the algorithm, try to find a simpler defining polynomial for the
intermediate number fields using PARI’s polred() . This usually speeds up the computation but can
also considerably slow it down. Try and see what works best in the given situation.
•simplify_all – (default: False ) If True , simplify intermediate fields and also the resulting number
field.
OUTPUT:
If map is False , the splitting field as an absolute number field. If map is True , a tuple (K,phi) where
phi is an embedding of the base field in K .
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = (x^3 + 2).splitting_field(); K
Number Field in a with defining polynomial x^6 + 3*x^5 + 6*x^4 + 11*x^3 + 12*x^2 ˓→ 3*x + 1
sage: K.<a> = (x^3 - 3*x + 1).splitting_field(); K
Number Field in a with defining polynomial x^3 - 3*x + 1
The simplify and simplify_all flags usually yield fields defined by polynomials with smaller coefficients. By default, simplify is True and simplify_all is False.
sage: (x^4 - x + 1).splitting_field('a', simplify=False)
Number Field in a with defining polynomial x^24 - 2780*x^22 + 2*x^21 + 3527512*x^
˓→20 - 2876*x^19 - 2701391985*x^18 + 945948*x^17 + 1390511639677*x^16 +
˓→736757420*x^15 - 506816498313560*x^14 - 822702898220*x^13 +
˓→134120588299548463*x^12 + 362240696528256*x^11 - 25964582366880639486*x^10 ˓→91743672243419990*x^9 + 3649429473447308439427*x^8 + 14310332927134072336*x^7 ˓→363192569823568746892571*x^6 - 1353403793640477725898*x^5 +
˓→24293393281774560140427565*x^4 + 70673814899934142357628*x^3 ˓→980621447508959243128437933*x^2 - 1539841440617805445432660*x +
˓→18065914012013502602456565991
sage: (x^4 - x + 1).splitting_field('a', simplify=True)
Number Field in a with defining polynomial x^24 + 8*x^23 - 32*x^22 - 310*x^21 +
˓→540*x^20 + 4688*x^19 - 6813*x^18 - 32380*x^17 + 49525*x^16 + 102460*x^15 ˓→129944*x^14 - 287884*x^13 + 372727*x^12 + 150624*x^11 - 110530*x^10 - 566926*x^
˓→9 + 1062759*x^8 - 779940*x^7 + 863493*x^6 - 1623578*x^5 + 1759513*x^4 ˓→955624*x^3 + 459975*x^2 - 141948*x + 53919
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20 + 4*x^19
˓→+ 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^12 - 15*x^
˓→11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^2 - 3*x +
˓→1
Reducible polynomials also work:
sage: pol = (x^4 - 1)*(x^2 + 1/2)*(x^2 + 1/3)
sage: pol.splitting_field('a', simplify_all=True)
Number Field in a with defining polynomial x^8 - x^4 + 1
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Relative situation:
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 + 2)
sage: S.<t> = PolynomialRing(K)
sage: L.<b> = (t^2 - a).splitting_field()
sage: L
Number Field in b with defining polynomial t^6 + 2
With map=True , we also get the embedding of the base field into the splitting field:
sage: L.<b>, phi = (t^2 - a).splitting_field(map=True)
sage: phi
Ring morphism:
From: Number Field in a with defining polynomial x^3 + 2
To:
Number Field in b with defining polynomial t^6 + 2
Defn: a |--> b^2
sage: (x^4 - x + 1).splitting_field('a', simplify_all=True, map=True)[1]
Ring morphism:
From: Rational Field
To:
Number Field in a with defining polynomial x^24 - 3*x^23 + 2*x^22 - x^20
˓→+ 4*x^19 + 32*x^18 - 35*x^17 - 92*x^16 + 49*x^15 + 163*x^14 - 15*x^13 - 194*x^
˓→12 - 15*x^11 + 163*x^10 + 49*x^9 - 92*x^8 - 35*x^7 + 32*x^6 + 4*x^5 - x^4 + 2*x^
˓→2 - 3*x + 1
Defn: 1 |--> 1
We can enable verbose messages:
sage: set_verbose(2)
sage: K.<a> = (x^3 - x + 1).splitting_field()
verbose 1 (...: splitting_field.py, splitting_field) Starting field: y
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor:
˓→[(3, 0)]
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle:
˓→[(2, 2), (3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree:
˓→[6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^2 + 23
verbose 1 (...: splitting_field.py, splitting_field) New field before simplifying:
˓→ x^2 + 23 (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) New field: y^2 - y + 6 (time
˓→= ...)
verbose 2 (...: splitting_field.py, splitting_field) Converted polynomials to new
˓→field (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to factor: []
verbose 2 (...: splitting_field.py, splitting_field) Done factoring (time = ...)
verbose 1 (...: splitting_field.py, splitting_field) SplittingData to handle:
˓→[(3, 3)]
verbose 1 (...: splitting_field.py, splitting_field) Bounds for absolute degree:
˓→[6, 6]
verbose 2 (...: splitting_field.py, splitting_field) Handling polynomial x^3 - x
˓→+ 1
verbose 1 (...: splitting_field.py, splitting_field) New field: y^6 + 3*y^5 +
˓→19*y^4 + 35*y^3 + 127*y^2 + 73*y + 271 (time = ...)
sage: set_verbose(0)
Try all Galois groups in degree 4. We use a quadratic base field such that polgalois() cannot be used:
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sage: R.<x> = PolynomialRing(QuadraticField(-11))
sage: C2C2pol = x^4 - 10*x^2 + 1
sage: C2C2pol.splitting_field('x')
Number Field in x with defining polynomial x^8 + 24*x^6 + 608*x^4 + 9792*x^2 +
˓→53824
sage: C4pol = x^4 + x^3 + x^2 + x + 1
sage: C4pol.splitting_field('x')
Number Field in x with defining polynomial x^8 - x^7 - 2*x^6 + 5*x^5 + x^4 + 15*x^
˓→3 - 18*x^2 - 27*x + 81
sage: D8pol = x^4 - 2
sage: D8pol.splitting_field('x')
Number Field in x with defining polynomial x^16 + 8*x^15 + 68*x^14 + 336*x^13 +
˓→1514*x^12 + 5080*x^11 + 14912*x^10 + 35048*x^9 + 64959*x^8 + 93416*x^7 +
˓→88216*x^6 + 41608*x^5 - 25586*x^4 - 60048*x^3 - 16628*x^2 + 12008*x + 34961
sage: A4pol = x^4 - 4*x^3 + 14*x^2 - 28*x + 21
sage: A4pol.splitting_field('x')
Number Field in x with defining polynomial x^24 - 20*x^23 + 290*x^22 - 3048*x^21
˓→+ 26147*x^20 - 186132*x^19 + 1130626*x^18 - 5913784*x^17 + 26899345*x^16 ˓→106792132*x^15 + 371066538*x^14 - 1127792656*x^13 + 2991524876*x^12 ˓→6888328132*x^11 + 13655960064*x^10 - 23000783036*x^9 + 32244796382*x^8 ˓→36347834476*x^7 + 30850889884*x^6 - 16707053128*x^5 + 1896946429*x^4 +
˓→4832907884*x^3 - 3038258802*x^2 - 200383596*x + 593179173
sage: S4pol = x^4 + x + 1
sage: S4pol.splitting_field('x')
Number Field in x with defining polynomial x^48 ...
Some bigger examples:
sage: R.<x> = PolynomialRing(QQ)
sage: pol15 = chebyshev_T(31, x) - 1
# 2^30*(x-1)*minpoly(cos(2*pi/31))^2
sage: pol15.splitting_field('a')
Number Field in a with defining polynomial x^15 - x^14 - 14*x^13 + 13*x^12 + 78*x^
˓→11 - 66*x^10 - 220*x^9 + 165*x^8 + 330*x^7 - 210*x^6 - 252*x^5 + 126*x^4 + 84*x^
˓→3 - 28*x^2 - 8*x + 1
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
sage: pol48.splitting_field('a')
Number Field in a with defining polynomial x^48 ...
If you somehow know the degree of the field in advance, you should add a degree_multiple argument.
This can speed up the computation, in particular for polynomials of degree >= 12 or for relative extensions:
sage: pol15.splitting_field('a', degree_multiple=15)
Number Field in a with defining polynomial x^15 + x^14 - 14*x^13 - 13*x^12 + 78*x^
˓→11 + 66*x^10 - 220*x^9 - 165*x^8 + 330*x^7 + 210*x^6 - 252*x^5 - 126*x^4 + 84*x^
˓→3 + 28*x^2 - 8*x - 1
A value for degree_multiple which isn’t actually a multiple of the absolute degree of the splitting field
can either result in a wrong answer or the following exception:
sage: pol48.splitting_field('a', degree_multiple=20)
Traceback (most recent call last):
...
ValueError: inconsistent degree_multiple in splitting_field()
Compute the Galois closure as the splitting field of the defining polynomial:
sage: R.<x> = PolynomialRing(QQ)
sage: pol48 = x^6 - 4*x^4 + 12*x^2 - 12
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sage: K.<a> = NumberField(pol48)
sage: L.<b> = pol48.change_ring(K).splitting_field()
sage: L
Number Field in b with defining polynomial x^48 ...
Try all Galois groups over Q in degree 5 except for 𝑆5 (the latter is infeasible with the current implementation):
sage: C5pol = x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: C5pol.splitting_field('x')
Number Field in x with defining polynomial x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1
sage: D10pol = x^5 - x^4 - 5*x^3 + 4*x^2 + 3*x - 1
sage: D10pol.splitting_field('x')
Number Field in x with defining polynomial x^10 - 28*x^8 + 216*x^6 - 681*x^4 +
˓→902*x^2 - 401
sage: AGL_1_5pol = x^5 - 2
sage: AGL_1_5pol.splitting_field('x')
Number Field in x with defining polynomial x^20 + 10*x^19 + 55*x^18 + 210*x^17 +
˓→595*x^16 + 1300*x^15 + 2250*x^14 + 3130*x^13 + 3585*x^12 + 3500*x^11 + 2965*x^
˓→10 + 2250*x^9 + 1625*x^8 + 1150*x^7 + 750*x^6 + 400*x^5 + 275*x^4 + 100*x^3 +
˓→75*x^2 + 25
sage: A5pol = x^5 - x^4 + 2*x^2 - 2*x + 2
sage: A5pol.splitting_field('x')
Number Field in x with defining polynomial x^60 ...
We can use the abort_degree option if we don’t want to compute fields of too large degree (this can be
used to check whether the splitting field has small degree):
sage: (x^5+x+3).splitting_field('b', abort_degree=119)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field equals 120
sage: (x^10+x+3).splitting_field('b', abort_degree=60) # long time (10s on sage.
˓→math, 2014)
Traceback (most recent call last):
...
SplittingFieldAbort: degree of splitting field is a multiple of 180
Use the degree_divisor attribute to recover the divisor of the degree of the splitting field or
degree_multiple to recover a multiple:
sage: from sage.rings.number_field.splitting_field import SplittingFieldAbort
sage: try: # long time (4s on sage.math, 2014)
....:
(x^8+x+1).splitting_field('b', abort_degree=60, simplify=False)
....: except SplittingFieldAbort as e:
....:
print(e.degree_divisor)
....:
print(e.degree_multiple)
120
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AUTHORS:
• William Stein (2004, 2005): initial version
• David Loeffler (2009): rewrite to give explicit homomorphism groups
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sage.rings.number_field.galois_group. GaloisGroup
alias of GaloisGroup_v1
class sage.rings.number_field.galois_group. GaloisGroupElement
Bases: sage.groups.perm_gps.permgroup_element.PermutationGroupElement
An element of a Galois group. This is stored as a permutation, but may also be made to act on elements of the
field (generally returning elements of its Galois closure).
EXAMPLES:
sage: K.<w> = QuadraticField(-7); G = K.galois_group()
sage: G[1]
(1,2)
sage: G[1](w + 2)
-w + 2
sage: L.<v> = NumberField(x^3 - 2); G = L.galois_group(names='y')
sage: G[4]
(1,5)(2,4)(3,6)
sage: G[4](v)
1/18*y^4
sage: G[4](G[4](v))
-1/36*y^4 - 1/2*y
sage: G[4](G[4](G[4](v)))
1/18*y^4
as_hom ( )
Return the homomorphism L -> L corresponding to self, where L is the Galois closure of the ambient
number field.
EXAMPLES:
sage: G = QuadraticField(-7,'w').galois_group()
sage: G[1].as_hom()
Ring endomorphism of Number Field in w with defining polynomial x^2 + 7
Defn: w |--> -w
ramification_degree ( P)
Return the greatest value of v such that s acts trivially modulo P^v. Should only be used if P is prime and
s is in the decomposition group of P.
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
4
K.<b> = NumberField(x^3 - 3, 'a').galois_closure()
G = K.galois_group()
P = K.primes_above(3)[0]
s = hom(K, K, 1/18*b^4 - 1/2*b)
G(s).ramification_degree(P)
class sage.rings.number_field.galois_group. GaloisGroup_subgroup ( ambient, elts)
Bases: sage.rings.number_field.galois_group.GaloisGroup_v2
A subgroup of a Galois group, as returned by functions such as decomposition_group .
fixed_field ( )
Return the fixed field of this subgroup (as a subfield of the Galois closure of the number field associated to
the ambient Galois group).
EXAMPLES:
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sage: L.<a> = NumberField(x^4 + 1)
sage: G = L.galois_group()
sage: H = G.decomposition_group(L.primes_above(3)[0])
sage: H.fixed_field()
(Number Field in a0 with defining polynomial x^2 + 2, Ring morphism:
From: Number Field in a0 with defining polynomial x^2 + 2
To:
Number Field in a with defining polynomial x^4 + 1
Defn: a0 |--> a^3 + a)
class sage.rings.number_field.galois_group. GaloisGroup_v1 ( group, number_field)
Bases: sage.structure.sage_object.SageObject
A wrapper around a class representing an abstract transitive group.
This is just a fairly minimal object at present. To get the underlying group, do G.group() , and to get the corresponding number field do G.number_field() . For a more sophisticated interface use the type=None
option.
EXAMPLES:
sage: K = QQ[2^(1/3)]
sage: G = K.galois_group(type="pari"); G
Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a
˓→with defining polynomial x^3 - 2
sage: G.order()
6
sage: G.group()
PARI group [6, -1, 2, "S3"] of degree 3
sage: G.number_field()
Number Field in a with defining polynomial x^3 - 2
group ( )
Return the underlying abstract group.
EXAMPLES:
sage: G = NumberField(x^3 + 2*x + 2, 'theta').galois_group(type="pari")
sage: H = G.group(); H
PARI group [6, -1, 2, "S3"] of degree 3
sage: P = H.permutation_group(); P # optional - database_gap
Transitive group number 2 of degree 3
sage: list(P)
# optional - database_gap
[(), (1,2), (1,2,3), (2,3), (1,3,2), (1,3)]
number_field ( )
Return the number field of which this is the Galois group.
EXAMPLES:
sage: G = NumberField(x^6 + 2, 't').galois_group(type="pari"); G
Galois group PARI group [12, -1, 3, "D(6) = S(3)[x]2"] of degree 6 of the
˓→Number Field in t with defining polynomial x^6 + 2
sage: G.number_field()
Number Field in t with defining polynomial x^6 + 2
order ( )
Return the order of this Galois group.
EXAMPLES:
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sage: G = NumberField(x^5 + 2, 'theta_1').galois_group(type="pari"); G
Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of degree 5 of the Number
˓→Field in theta_1 with defining polynomial x^5 + 2
sage: G.order()
20
class sage.rings.number_field.galois_group. GaloisGroup_v2 ( number_field,
names=None)
Bases: sage.groups.perm_gps.permgroup.PermutationGroup_generic
The Galois group of an (absolute) number field.
Note: We define the Galois group of a non-normal field K to be the Galois group of its Galois closure L, and
elements are stored as permutations of the roots of the defining polynomial of L, not as permutations of the roots
(in L) of the defining polynomial of K. The latter would probably be preferable, but is harder to implement. Thus
the permutation group that is returned is always simply-transitive.
The ‘arithmetical’ features (decomposition and ramification groups, Artin symbols etc) are only available for
Galois fields.
artin_symbol ( P)
(︁
)︁
Return the Artin symbol 𝐾/Q
, where K is the number field of self, and P is an unramified prime ideal.
P
This is the unique element s of the decomposition group of P such that 𝑠(𝑥) = 𝑥𝑝 mod P, where p is the
residue characteristic of P.
EXAMPLES:
sage: K.<b> = NumberField(x^4 - 2*x^2 + 2, 'a').galois_closure()
sage: G = K.galois_group()
sage: [G.artin_symbol(P) for P in K.primes_above(7)]
[(1,5)(2,6)(3,7)(4,8), (1,5)(2,6)(3,7)(4,8), (1,4)(2,3)(5,8)(6,7),
˓→(1,4)(2,3)(5,8)(6,7)]
sage: G.artin_symbol(17)
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not prime
sage: QuadraticField(-7,'c').galois_group().artin_symbol(13)
(1,2)
sage: G.artin_symbol(K.primes_above(2)[0])
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is ramified
complex_conjugation ( P=None)
Return the unique element of self corresponding to complex conjugation, for a specified embedding P into
the complex numbers. If P is not specified, use the “standard” embedding, whenever that is well-defined.
EXAMPLES:
sage: L.<z> = CyclotomicField(7)
sage: G = L.galois_group()
sage: conj = G.complex_conjugation(); conj
(1,4)(2,5)(3,6)
sage: conj(z)
-z^5 - z^4 - z^3 - z^2 - z - 1
An example where the field is not CM, so complex conjugation really depends on the choice of embedding:
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sage: L = NumberField(x^6 + 40*x^3 + 1372,'a')
sage: G = L.galois_group()
sage: [G.complex_conjugation(x) for x in L.places()]
[(1,3)(2,6)(4,5), (1,5)(2,4)(3,6), (1,2)(3,4)(5,6)]
decomposition_group ( P)
Decomposition group of a prime ideal P, i.e. the subgroup of elements that map P to itself. This is the
same as the Galois group of the extension of local fields obtained by completing at P.
This function will raise an error if P is not prime or the given number field is not Galois.
P can also be an infinite prime, i.e. an embedding into R or C.
EXAMPLES:
sage: K.<a> = NumberField(x^4 - 2*x^2 + 2,'b').galois_closure()
sage: P = K.ideal([17, a^2])
sage: G = K.galois_group()
sage: G.decomposition_group(P)
Subgroup [(), (1,8)(2,7)(3,6)(4,5)] of Galois group of Number Field in a with
˓→defining polynomial x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156
sage: G.decomposition_group(P^2)
Traceback (most recent call last):
...
ValueError: Fractional ideal (...) is not prime
sage: G.decomposition_group(17)
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not prime
An example with an infinite place:
sage: L.<b> = NumberField(x^3 - 2,'a').galois_closure(); G=L.galois_group()
sage: x = L.places()[0]
sage: G.decomposition_group(x).order()
2
inertia_group ( P)
Return the inertia group of the prime P, i.e. the group of elements acting trivially modulo P. This is just the
0th ramification group of P.
EXAMPLES:
sage: K.<b> = NumberField(x^2 - 3,'a')
sage: G = K.galois_group()
sage: G.inertia_group(K.primes_above(2)[0])
Galois group of Number Field in b with defining polynomial x^2 - 3
sage: G.inertia_group(K.primes_above(5)[0])
Subgroup [()] of Galois group of Number Field in b with defining polynomial x^
˓→2 - 3
is_galois ( )
Return True if the underlying number field of self is actually Galois.
EXAMPLES:
sage: NumberField(x^3 - x + 1,'a').galois_group(names='b').is_galois()
False
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sage: NumberField(x^2 - x + 1,'a').galois_group().is_galois()
True
list ( )
List of the elements of self.
EXAMPLES:
sage: NumberField(x^3 - 3*x + 1,'a').galois_group().list()
[(), (1,2,3), (1,3,2)]
ngens ( )
Number of generators of self.
EXAMPLES:
sage: QuadraticField(-23, 'a').galois_group().ngens()
1
number_field ( )
The ambient number field.
EXAMPLES:
sage: K = NumberField(x^3 - x + 1, 'a')
sage: K.galois_group(names='b').number_field() is K
True
ramification_breaks ( P)
Return the set of ramification breaks of the prime ideal P, i.e. the set of indices i such that the ramification
group 𝐺𝑖+1 ̸= 𝐺𝑖 . This is only defined for Galois fields.
EXAMPLES:
sage: K.<b> = NumberField(x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156)
sage: G = K.galois_group()
sage: P = K.primes_above(2)[0]
sage: G.ramification_breaks(P)
{1, 3, 5}
sage: min( [ G.ramification_group(P, i).order() / G.ramification_group(P,
˓→i+1).order() for i in G.ramification_breaks(P)] )
2
ramification_group ( P, v)
Return the vth ramification group of self for the prime P, i.e. the set of elements s of self such that s acts
trivially modulo P^(v+1). This is only defined for Galois fields.
EXAMPLES:
sage: K.<b> = NumberField(x^3 - 3,'a').galois_closure()
sage: G=K.galois_group()
sage: P = K.primes_above(3)[0]
sage: G.ramification_group(P, 3)
Subgroup [(), (1,2,4)(3,5,6), (1,4,2)(3,6,5)] of Galois group of Number Field
˓→in b with defining polynomial x^6 + 243
sage: G.ramification_group(P, 5)
Subgroup [()] of Galois group of Number Field in b with defining polynomial x^
˓→6 + 243
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splitting_field ( )
The Galois closure of the ambient number field.
EXAMPLES:
sage: K = NumberField(x^3 - x + 1, 'a')
sage: K.galois_group(names='b').splitting_field()
Number Field in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23
sage: L = QuadraticField(-23, 'c'); L.galois_group().splitting_field() is L
True
subgroup ( elts)
Return the subgroup of self with the given elements. Mostly for internal use.
EXAMPLES:
sage: G = NumberField(x^3 - x - 1, 'a').galois_closure('b').galois_group()
sage: G.subgroup([ G(1), G([(1,2,3),(4,5,6)]), G([(1,3,2),(4,6,5)]) ])
Subgroup [(), (1,2,3)(4,5,6), (1,3,2)(4,6,5)] of Galois group of Number Field
˓→in b with defining polynomial x^6 - 6*x^4 + 9*x^2 + 23
unrank ( i)
Return the i -th element of self .
INPUT:
•i – integer between 0 and n-1 where n is the cardinality of this set
EXAMPLES:
sage: G = NumberField(x^3 - 3*x + 1,'a').galois_group()
sage: [G.unrank(i) for i in range(G.cardinality())]
[(), (1,2,3), (1,3,2)]
1.8 Elements of bounded height in number fields
Sage functions to list all elements of a given number field with height less than a specified bound.
AUTHORS:
• John Doyle (2013): initial version
• David Krumm (2013): initial version
REFERENCES:
sage.rings.number_field.bdd_height. bdd_height ( K,
height_bound,
precision=53,
LLL=False)
Computes all elements in the number number field 𝐾 which have relative multiplicative height at most
height_bound .
The algorithm requires arithmetic with floating point numbers; precision gives the user the option to set the
precision for such computations.
It might be helpful to work with an LLL-reduced system of fundamental units, so the user has the option to
perform an LLL reduction for the fundamental units by setting LLL to True.
Certain computations may be faster assuming GRH, which may be done globally by using the number_field(True/False) switch.
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The function will only be called for number fields 𝐾 with positive unit rank. An error will occur if 𝐾 is 𝑄𝑄 or
an imaginary quadratic field.
ALGORITHM:
This is an implementation of the main algorithm (Algorithm 3) in [Doyle-Krumm].
INPUT:
•height_bound - real number
•precision - (default: 53) positive integer
•LLL - (default: False) boolean value
OUTPUT:
•an iterator of number field elements
Warning: In the current implementation, the output of the algorithm cannot be guaranteed to be correct due
to the necessity of floating point computations. In some cases, the default 53-bit precision is considerably
lower than would be required for the algorithm to generate correct output.
Todo
Should implement a version of the algorithm that guarantees correct output. See Algorithm 4 in [Doyle-Krumm]
for details of an implementation that takes precision issues into account.
EXAMPLES:
There are no elements of negative height:
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = NumberField(x^5 - x + 7)
sage: list(bdd_height(K,-3))
[]
The only nonzero elements of height 1 are the roots of unity:
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = QuadraticField(3)
sage: list(bdd_height(K,1))
[0, -1, 1]
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = QuadraticField(36865)
sage: len(list(bdd_height(K,101))) # long time (4 s)
131
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = NumberField(x^3 - 197*x + 39)
sage: len(list(bdd_height(K, 200))) # long time (5 s)
451
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = NumberField(x^6 + 2)
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sage: len(list(bdd_height(K,60,precision=100))) # long time (5 s)
1899
sage: from sage.rings.number_field.bdd_height import bdd_height
sage: K.<g> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
sage: len(list(bdd_height(K,10,LLL=true)))
99
sage.rings.number_field.bdd_height. bdd_height_iq ( K, height_bound)
Compute all elements in the imaginary quadratic field 𝐾 which have relative multiplicative height at most
height_bound .
The function will only be called with 𝐾 an imaginary quadratic field.
If called with 𝐾 not an imaginary quadratic, the function will likely yield incorrect output.
ALGORITHM:
This is an implementation of Algorithm 5 in [Doyle-Krumm].
INPUT:
•𝐾 - an imaginary quadratic number field
•height_bound - a real number
OUTPUT:
•an iterator of number field elements
EXAMPLES:
sage: from sage.rings.number_field.bdd_height import bdd_height_iq
sage: K.<a> = NumberField(x^2 + 191)
sage: for t in bdd_height_iq(K,8):
....:
print(exp(2*t.global_height()))
1.00000000000000
1.00000000000000
1.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000
4.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
8.00000000000000
√︀
There are 175 elements of height at most 10 in 𝑄𝑄( ( − 3)):
sage: from sage.rings.number_field.bdd_height import bdd_height_iq
sage: K.<a> = NumberField(x^2 + 3)
sage: len(list(bdd_height_iq(K,10)))
175
The only elements of multiplicative height 1 in a number field are 0 and the roots of unity:
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sage:
sage:
sage:
[0, a
from sage.rings.number_field.bdd_height import bdd_height_iq
K.<a> = NumberField(x^2 + x + 1)
list(bdd_height_iq(K,1))
+ 1, a, -1, -a - 1, -a, 1]
A number field has no elements of multiplicative height less than 1:
sage: from sage.rings.number_field.bdd_height import bdd_height_iq
sage: K.<a> = NumberField(x^2 + 5)
sage: list(bdd_height_iq(K,0.9))
[]
sage.rings.number_field.bdd_height. bdd_norm_pr_gens_iq ( K, norm_list)
Compute generators for all principal ideals in an imaginary quadratic field 𝐾 whose norms are in norm_list
.
The only keys for the output dictionary are integers n appearing in norm_list .
The function will only be called with 𝐾 an imaginary quadratic field.
The function will return a dictionary for other number fields, but it may be incorrect.
INPUT:
•𝐾 - an imaginary quadratic number field
•norm_list - a list of positive integers
OUTPUT:
•a dictionary of number field elements, keyed by norm
EXAMPLES:
In 𝑄𝑄(𝑖), there is one principal ideal of norm 4, two principal ideals of norm 5, but no principal ideals of norm
7:
sage:
sage:
sage:
sage:
sage:
[2]
sage:
[-g sage:
[]
from sage.rings.number_field.bdd_height import bdd_norm_pr_gens_iq
K.<g> = NumberField(x^2 + 1)
L = range(10)
bdd_pr_ideals = bdd_norm_pr_gens_iq(K, L)
bdd_pr_ideals[4]
bdd_pr_ideals[5]
2, -g + 2]
bdd_pr_ideals[7]
There are no ideals in the ring of integers with negative norm:
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_gens_iq
sage: K.<g> = NumberField(x^2 + 10)
sage: L = range(-5,-1)
sage: bdd_pr_ideals = bdd_norm_pr_gens_iq(K,L)
sage: bdd_pr_ideals
{-5: [], -4: [], -3: [], -2: []}
Calling a key that is not in the input norm_list raises a KeyError:
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_gens_iq
sage: K.<g> = NumberField(x^2 + 20)
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sage: L = range(100)
sage: bdd_pr_ideals = bdd_norm_pr_gens_iq(K, L)
sage: bdd_pr_ideals[100]
Traceback (most recent call last):
...
KeyError: 100
sage.rings.number_field.bdd_height. bdd_norm_pr_ideal_gens ( K, norm_list)
Compute generators for all principal ideals in a number field 𝐾 whose norms are in norm_list .
INPUT:
•𝐾 - a number field
•norm_list - a list of positive integers
OUTPUT:
•a dictionary of number field elements, keyed by norm
EXAMPLES:
There is only one principal ideal of norm 1, and it is generated by the element 1:
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_ideal_gens
sage: K.<g> = QuadraticField(101)
sage: bdd_norm_pr_ideal_gens(K, [1])
{1: [1]}
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_ideal_gens
sage: K.<g> = QuadraticField(123)
sage: bdd_norm_pr_ideal_gens(K, range(5))
{0: [0], 1: [1], 2: [-g - 11], 3: [], 4: [2]}
sage: from sage.rings.number_field.bdd_height import bdd_norm_pr_ideal_gens
sage: K.<g> = NumberField(x^5 - x + 19)
sage: b = bdd_norm_pr_ideal_gens(K, range(30))
sage: key = ZZ(28)
sage: b[key]
[157*g^4 - 139*g^3 - 369*g^2 + 848*g + 158, g^4 + g^3 - g - 7]
sage.rings.number_field.bdd_height. integer_points_in_polytope ( matrix, interval_radius)
Return the set of integer points in the polytope obtained by acting on a cube by a linear transformation.
Given an r-by-r matrix matrix and a real number interval_radius , this function finds all integer lattice
points in the polytope obtained by transforming the cube [-interval_radius,interval_radius]^r via the linear map
induced by matrix .
INPUT:
•matrix - a square matrix of real numbers
•interval_radius - a real number
OUTPUT:
•a list of tuples of integers
EXAMPLES:
Stretch the interval [-1,1] by a factor of 2 and find the integers in the resulting interval:
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sage: from sage.rings.number_field.bdd_height import integer_points_in_polytope
sage: m = matrix([2])
sage: r = 1
sage: integer_points_in_polytope(m,r)
[(-2), (-1), (0), (1), (2)]
Integer points inside a parallelogram:
sage:
sage:
sage:
sage:
[(-3,
˓→1),
from sage.rings.number_field.bdd_height import integer_points_in_polytope
m = matrix([[1, 2],[3, 4]])
r = RealField()(1.3)
integer_points_in_polytope(m,r)
-7), (-2, -5), (-2, -4), (-1, -3), (-1, -2), (-1, -1), (0, -1), (0, 0), (0,
(1, 1), (1, 2), (1, 3), (2, 4), (2, 5), (3, 7)]
Integer points inside a parallelepiped:
sage:
sage:
sage:
sage:
sage:
4143
from sage.rings.number_field.bdd_height import integer_points_in_polytope
m = matrix([[1.2,3.7,0.2],[-5.3,-.43,3],[1.2,4.7,-2.1]])
r = 2.2
L = integer_points_in_polytope(m,r)
len(L)
If interval_radius is 0, the output should include only the zero tuple:
sage: from sage.rings.number_field.bdd_height import integer_points_in_polytope
sage: m = matrix([[1,2,3,7],[4,5,6,2],[7,8,9,3],[0,3,4,5]])
sage: integer_points_in_polytope(m,0)
[(0, 0, 0, 0)]
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CHAPTER
TWO
MORPHISMS
2.1 Morphisms between number fields
This module provides classes to represent ring homomorphisms between number fields (i.e. field embeddings).
class sage.rings.number_field.morphism. CyclotomicFieldHomomorphism_im_gens
Bases: sage.rings.number_field.morphism.NumberFieldHomomorphism_im_gens
class sage.rings.number_field.morphism. CyclotomicFieldHomset ( R,
S,
gory=None)
Bases: sage.rings.number_field.morphism.NumberFieldHomset
cate-
Set of homomorphisms with domain a given cyclotomic field.
EXAMPLES:
sage: End(CyclotomicField(16))
Automorphism group of Cyclotomic Field of order 16 and degree 8
list ( )
Return a list of all the elements of self (for which the domain is a cyclotomic field).
EXAMPLES:
sage: K.<z> = CyclotomicField(12)
sage: G = End(K); G
Automorphism group of Cyclotomic Field of order 12 and degree 4
sage: [g(z) for g in G]
[z, z^3 - z, -z, -z^3 + z]
sage: L.<a, b> = NumberField([x^2 + x + 1, x^4 + 1])
sage: L
Number Field in a with defining polynomial x^2 + x + 1 over its base field
sage: Hom(CyclotomicField(12), L)[3]
Ring morphism:
From: Cyclotomic Field of order 12 and degree 4
To:
Number Field in a with defining polynomial x^2 + x + 1 over its base
˓→field
Defn: zeta12 |--> -b^2*a
sage: list(Hom(CyclotomicField(5), K))
[]
sage: Hom(CyclotomicField(11), L).list()
[]
class sage.rings.number_field.morphism. NumberFieldHomomorphism_im_gens
Bases: sage.rings.morphism.RingHomomorphism_im_gens
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preimage ( y)
Computes a preimage of 𝑦 in the domain, provided one exists. Raises a ValueError if 𝑦 has no preimage.
INPUT:
•𝑦 – an element of the codomain of self.
OUTPUT:
Returns the preimage of 𝑦 in the domain, if one exists. Raises a ValueError if 𝑦 has no preimage.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 7)
sage: L.<b> = NumberField(x^4 - 7)
sage: f = K.embeddings(L)[0]
sage: f.preimage(3*b^2 - 12/7)
3*a - 12/7
sage: f.preimage(b)
Traceback (most recent call last):
...
ValueError: Element 'b' is not in the image of this homomorphism.
sage:
sage:
sage:
sage:
2*b -
F.<b> = QuadraticField(23)
G.<a> = F.extension(x^3+5)
f = F.embeddings(G)[0]
f.preimage(a^3+2*b+3)
2
class sage.rings.number_field.morphism. NumberFieldHomset ( R, S, category=None)
Bases: sage.rings.homset.RingHomset_generic
Set of homomorphisms with domain a given number field.
cardinality ( )
Return the order of this set of field homomorphism.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 1)
sage: End(k)
Automorphism group of Number Field in a with defining polynomial x^2 + 1
sage: End(k).order()
2
sage: k.<a> = NumberField(x^3 + 2)
sage: End(k).order()
1
sage: K.<a> = NumberField( [x^3 + 2, x^2 + x + 1] )
sage: End(K).order()
6
list ( )
Return a list of all the elements of self.
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 3*x + 1)
sage: End(K).list()
[
Ring endomorphism of Number Field in a with defining polynomial x^3 - 3*x + 1
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Defn: a |--> a,
Ring endomorphism of Number Field in a with defining polynomial x^3 - 3*x + 1
Defn: a |--> a^2 - 2,
Ring endomorphism of Number Field in a with defining polynomial x^3 - 3*x + 1
Defn: a |--> -a^2 - a + 2
]
sage: Hom(K, CyclotomicField(9))[0] # indirect doctest
Ring morphism:
From: Number Field in a with defining polynomial x^3 - 3*x + 1
To:
Cyclotomic Field of order 9 and degree 6
Defn: a |--> -zeta9^4 + zeta9^2 - zeta9
An example where the codomain is a relative extension:
sage: K.<a> = NumberField(x^3 - 2)
sage: L.<b> = K.extension(x^2 + 3)
sage: Hom(K, L).list()
[
Ring morphism:
From: Number Field in a with defining
To:
Number Field in b with defining
Defn: a |--> a,
Ring morphism:
From: Number Field in a with defining
To:
Number Field in b with defining
Defn: a |--> -1/2*a*b - 1/2*a,
Ring morphism:
From: Number Field in a with defining
To:
Number Field in b with defining
Defn: a |--> 1/2*a*b - 1/2*a
]
polynomial x^3 - 2
polynomial x^2 + 3 over its base field
polynomial x^3 - 2
polynomial x^2 + 3 over its base field
polynomial x^3 - 2
polynomial x^2 + 3 over its base field
order ( )
Return the order of this set of field homomorphism.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 1)
sage: End(k)
Automorphism group of Number Field in a with defining polynomial x^2 + 1
sage: End(k).order()
2
sage: k.<a> = NumberField(x^3 + 2)
sage: End(k).order()
1
sage: K.<a> = NumberField( [x^3 + 2, x^2 + x + 1] )
sage: End(K).order()
6
class sage.rings.number_field.morphism. RelativeNumberFieldHomomorphism_from_abs ( parent,
abs_hom)
Bases: sage.rings.morphism.RingHomomorphism
A homomorphism from a relative number field to some other ring, stored as a homomorphism from the corresponding absolute field.
abs_hom ( )
Return the corresponding homomorphism from the absolute number field.
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EXAMPLES:
sage: K.<a, b>
sage: K.hom(a,
Ring morphism:
From: Number
˓→3 - 9*x + 9
To:
Number
Defn: a |-->
= NumberField( [x^3 + 2, x^2 + x + 1] )
K).abs_hom()
Field in a with defining polynomial x^6 - 3*x^5 + 6*x^4 - 3*x^
Field in a with defining polynomial x^3 + 2 over its base field
a - b
im_gens ( )
Return the images of the generators under this map.
EXAMPLES:
sage: K.<a, b> = NumberField( [x^3 + 2, x^2 + x + 1] )
sage: K.hom(a, K).im_gens()
[a, b]
class sage.rings.number_field.morphism. RelativeNumberFieldHomset ( R, S, category=None)
Bases: sage.rings.number_field.morphism.NumberFieldHomset
Set of homomorphisms with domain a given relative number field.
EXAMPLES:
We construct a homomorphism from a relative field by giving the image of a generator:
sage: L.<cuberoot2, zeta3> = CyclotomicField(3).extension(x^3 - 2)
sage: phi = L.hom([cuberoot2 * zeta3]); phi
Relative number field endomorphism of Number Field in cuberoot2 with defining
˓→polynomial x^3 - 2 over its base field
Defn: cuberoot2 |--> zeta3*cuberoot2
zeta3 |--> zeta3
sage: phi(cuberoot2 + zeta3)
zeta3*cuberoot2 + zeta3
In fact, this phi is a generator for the Kummer Galois group of this cyclic extension:
sage: phi(phi(cuberoot2 + zeta3))
(-zeta3 - 1)*cuberoot2 + zeta3
sage: phi(phi(phi(cuberoot2 + zeta3)))
cuberoot2 + zeta3
default_base_hom ( )
Pick an embedding of the base field of self into the codomain of this homset. This is done in an essentially
arbitrary way.
EXAMPLES:
sage: L.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])
sage: M.<c> = NumberField(x^4 + 80*x^2 + 36)
sage: Hom(L, M).default_base_hom()
Ring morphism:
From: Number Field in b with defining polynomial x^2 + 23
To:
Number Field in c with defining polynomial x^4 + 80*x^2 + 36
Defn: b |--> 1/12*c^3 + 43/6*c
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list ( )
Return a list of all the elements of self (for which the domain is a relative number field).
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + x + 1, x^3 + 2])
sage: End(K).list()
[
Relative number field endomorphism of Number Field in a with defining
˓→polynomial x^2 + x + 1 over its base field
Defn: a |--> a
b |--> b,
...
Relative number field endomorphism of Number Field in a with defining
˓→polynomial x^2 + x + 1 over its base field
Defn: a |--> a
b |--> -b*a - b
]
An example with an absolute codomain:
sage: K.<a, b> = NumberField([x^2 - 3, x^2 + 2])
sage: Hom(K, CyclotomicField(24, 'z')).list()
[
Relative number field morphism:
From: Number Field in a with defining polynomial x^2 - 3 over its base field
To:
Cyclotomic Field of order 24 and degree 8
Defn: a |--> z^6 - 2*z^2
b |--> -z^5 - z^3 + z,
...
Relative number field morphism:
From: Number Field in a with defining polynomial x^2 - 3 over its base field
To:
Cyclotomic Field of order 24 and degree 8
Defn: a |--> -z^6 + 2*z^2
b |--> z^5 + z^3 - z
]
2.2 Embeddings into ambient fields
This module provides classes to handle embeddings of number fields into ambient fields (generally R or C).
class sage.rings.number_field.number_field_morphisms. CyclotomicFieldEmbedding
Bases: sage.rings.number_field.number_field_morphisms.NumberFieldEmbedding
Specialized class for converting cyclotomic field elements into a cyclotomic field of higher order. All the real
work is done by _lift_cyclotomic_element.
class sage.rings.number_field.number_field_morphisms. EmbeddedNumberFieldConversion
Bases: sage.categories.map.Map
This allows one to cast one number field in another consistently, assuming they both have specified embeddings
into an ambient field (by default it looks for an embedding into C).
This is done by factoring the minimal polynomial of the input in the number field of the codomain. This may
fail if the element is not actually in the given field.
ambient_field
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class sage.rings.number_field.number_field_morphisms. EmbeddedNumberFieldMorphism
Bases: sage.rings.number_field.number_field_morphisms.NumberFieldEmbedding
This allows one to go from one number field in another consistently, assuming they both have specified embeddings into an ambient field.
If no ambient field is supplied, then the following ambient fields are tried:
•the pushout of the fields where the number fields are embedded;
•the algebraic closure of the previous pushout;
•C.
EXAMPLES:
sage: K.<i> = NumberField(x^2+1,embedding=QQbar(I))
sage: L.<i> = NumberField(x^2+1,embedding=-QQbar(I))
sage: from sage.rings.number_field.number_field_morphisms import
˓→EmbeddedNumberFieldMorphism
sage: EmbeddedNumberFieldMorphism(K,L,CDF)
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To:
Number Field in i with defining polynomial x^2 + 1
Defn: i -> -i
sage: EmbeddedNumberFieldMorphism(K,L,QQbar)
Generic morphism:
From: Number Field in i with defining polynomial x^2 + 1
To:
Number Field in i with defining polynomial x^2 + 1
Defn: i -> -i
ambient_field
section ( )
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import
˓→EmbeddedNumberFieldMorphism
sage: K.<a> = NumberField(x^2-700, embedding=25)
sage: L.<b> = NumberField(x^6-700, embedding=3)
sage: f = EmbeddedNumberFieldMorphism(K, L)
sage: f(2*a-1)
2*b^3 - 1
sage: g = f.section()
sage: g(2*b^3-1)
2*a - 1
class sage.rings.number_field.number_field_morphisms. NumberFieldEmbedding
Bases: sage.categories.morphism.Morphism
If R is a lazy field, the closest root to gen_embedding will be chosen.
EXAMPLES:
sage: x = polygen(QQ)
sage: from sage.rings.number_field.number_field_morphisms import
˓→NumberFieldEmbedding
sage: K.<a> = NumberField(x^3-2)
sage: f = NumberFieldEmbedding(K, RLF, 1)
sage: f(a)^3
2.00000000000000?
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sage: RealField(200)(f(a)^3)
2.0000000000000000000000000000000000000000000000000000000000
sage: sigma_a = K.polynomial().change_ring(CC).roots()[1][0]; sigma_a
-0.62996052494743... - 1.09112363597172*I
sage: g = NumberFieldEmbedding(K, CC, sigma_a)
sage: g(a+1)
0.37003947505256... - 1.09112363597172*I
gen_image ( )
Returns the image of the generator under this embedding.
EXAMPLES:
sage: f = QuadraticField(7, 'a', embedding=2).coerce_embedding()
sage: f.gen_image()
2.645751311064591?
sage.rings.number_field.number_field_morphisms. closest ( target, values, margin=1)
This is a utility function that returns the item in values closest to target (with respect to the code{abs} function).
If margin is greater than 1, and x and y are the first and second closest elements to target, then only return x if x
is margin times closer to target than y, i.e. margin * abs(target-x) < abs(target-y).
sage.rings.number_field.number_field_morphisms. create_embedding_from_approx ( K,
gen_image)
Return an embedding of K determined by gen_image .
The codomain of the embedding is the parent of gen_image or, if gen_image is not already an exact root
of the defining polynomial of K , the corresponding lazy field. The embedding maps the generator of K to a root
of the defining polynomial of K closest to gen_image .
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import create_embedding_
˓→from_approx
sage: K.<a> = NumberField(x^3-x+1/10)
sage: create_embedding_from_approx(K, 1)
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 1/10
To:
Real Lazy Field
Defn: a -> 0.9456492739235915?
sage: create_embedding_from_approx(K, 0)
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 1/10
To:
Real Lazy Field
Defn: a -> 0.10103125788101081?
sage: create_embedding_from_approx(K, -1)
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 1/10
To:
Real Lazy Field
Defn: a -> -1.046680531804603?
We can define embeddings from one number field to another:
sage: L.<b> = NumberField(x^6-x^2+1/10)
sage: create_embedding_from_approx(K, b^2)
Generic morphism:
From: Number Field in a with defining polynomial x^3 - x + 1/10
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To:
Number Field in b with defining polynomial x^6 - x^2 + 1/10
Defn: a -> b^2
If the embedding is exact, it must be valid:
sage: create_embedding_from_approx(K, b)
Traceback (most recent call last):
...
ValueError: b is not a root of x^3 - x + 1/10
sage.rings.number_field.number_field_morphisms. matching_root ( poly,
target,
ambient_field=None,
margin=1,
max_prec=None)
Given a polynomial and a target, this function chooses the root that target best approximates as compared in
ambient_field.
If the parent of target is exact, the equality is required, otherwise find closest root (with respect to the code{abs}
function) in the ambient field to the target, and return the root of poly (if any) that approximates it best.
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import matching_root
sage: R.<x> = CC[]
sage: matching_root(x^2-2, 1.5)
1.41421356237310
sage: matching_root(x^2-2, -100.0)
-1.41421356237310
sage: matching_root(x^2-2, .00000001)
1.41421356237310
sage: matching_root(x^3-1, CDF.0)
-0.50000000000000... + 0.86602540378443...*I
sage: matching_root(x^3-x, 2, ambient_field=RR)
1.00000000000000
sage.rings.number_field.number_field_morphisms. root_from_approx ( f, a)
Return an exact root of the polynomial 𝑓 closest to 𝑎.
INPUT:
•f – polynomial with rational coefficients
•a – element of a ring
OUTPUT:
A root of f in the parent of a or, if a is not already an exact root of f , in the corresponding lazy field. The
root is taken to be closest to a among all roots of f .
EXAMPLES:
sage: from sage.rings.number_field.number_field_morphisms import root_from_approx
sage: R.<x> = QQ[]
sage: root_from_approx(x^2 - 1, -1)
-1
sage: root_from_approx(x^2 - 2, 1)
1.414213562373095?
sage: root_from_approx(x^3 - x - 1, RR(1))
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1.324717957244746?
sage: root_from_approx(x^3 - x - 1, CC.gen())
-0.6623589786223730? + 0.5622795120623013?*I
sage: root_from_approx(x^2 + 1, 0)
Traceback (most recent call last):
...
ValueError: x^2 + 1 has no real roots
sage: root_from_approx(x^2 + 1, CC(0))
-1*I
sage: root_from_approx(x^2 - 2, sqrt(2))
sqrt(2)
sage: root_from_approx(x^2 - 2, sqrt(3))
Traceback (most recent call last):
...
ValueError: sqrt(3) is not a root of x^2 - 2
2.3 Structure maps for number fields
Provides isomorphisms between relative and absolute presentations, to and from vector spaces, name changing maps,
etc.
EXAMPLES:
sage: L.<cuberoot2, zeta3> = CyclotomicField(3).extension(x^3 - 2)
sage: K = L.absolute_field('a')
sage: from_K, to_K = K.structure()
sage: from_K
Isomorphism map:
From: Number Field in a with defining polynomial x^6 - 3*x^5 + 6*x^4 - 11*x^3 +
˓→12*x^2 + 3*x + 1
To:
Number Field in cuberoot2 with defining polynomial x^3 - 2 over its base field
sage: to_K
Isomorphism map:
From: Number Field in cuberoot2 with defining polynomial x^3 - 2 over its base field
To:
Number Field in a with defining polynomial x^6 - 3*x^5 + 6*x^4 - 11*x^3 +
˓→12*x^2 + 3*x + 1
class sage.rings.number_field.maps. MapAbsoluteToRelativeNumberField ( A, R)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
See MapRelativeToAbsoluteNumberField for examples.
class sage.rings.number_field.maps. MapNumberFieldToVectorSpace ( K, V)
Bases: sage.categories.map.Map
A class for the isomorphism from an absolute number field to its underlying Q-vector space.
EXAMPLES:
sage: L.<a> = NumberField(x^3 - x + 1)
sage: V, fr, to = L.vector_space()
sage: type(to)
<class 'sage.rings.number_field.maps.MapNumberFieldToVectorSpace'>
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class sage.rings.number_field.maps. MapRelativeNumberFieldToRelativeVectorSpace ( K,
V)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
EXAMPLES:
sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])
sage: V, fr, to = K.relative_vector_space()
sage: type(to)
<class 'sage.rings.number_field.maps.MapRelativeNumberFieldToRelativeVectorSpace'>
class sage.rings.number_field.maps. MapRelativeNumberFieldToVectorSpace ( L, V,
to_K,
to_V)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
The isomorphism from a relative number field to its underlying Q-vector space.
MapRelativeNumberFieldToRelativeVectorSpace .
Compare
EXAMPLES:
sage: K.<a> = NumberField(x^8 + 100*x^6 + x^2 + 5)
sage: L = K.relativize(K.subfields(4)[0][1], 'b'); L
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: L_to_K, K_to_L = L.structure()
sage: V, fr, to = L.absolute_vector_space()
sage: V
Vector space of dimension 8 over Rational Field
sage: fr
Isomorphism map:
From: Vector space of dimension 8 over Rational Field
To:
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: to
Isomorphism map:
From: Number Field in b with defining polynomial x^2 + a0 over its base field
To:
Vector space of dimension 8 over Rational Field
sage: type(fr), type(to)
(<class 'sage.rings.number_field.maps.MapVectorSpaceToRelativeNumberField'>,
<class 'sage.rings.number_field.maps.MapRelativeNumberFieldToVectorSpace'>)
sage: v = V([1, 1, 1, 1, 0, 1, 1, 1])
sage: fr(v), to(fr(v)) == v
((-a0^3 + a0^2 - a0 + 1)*b - a0^3 - a0 + 1, True)
sage: to(L.gen()), fr(to(L.gen())) == L.gen()
((0, 1, 0, 0, 0, 0, 0, 0), True)
class sage.rings.number_field.maps. MapRelativeToAbsoluteNumberField ( R, A)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
EXAMPLES:
sage: K.<a> = NumberField(x^6 + 4*x^2 + 200)
sage: L = K.relativize(K.subfields(3)[0][1], 'b'); L
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: fr, to = L.structure()
sage: fr
Relative number field morphism:
From: Number Field in b with defining polynomial x^2 + a0 over its base field
To:
Number Field in a with defining polynomial x^6 + 4*x^2 + 200
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Defn: b |--> a
a0 |--> -a^2
sage: to
Ring morphism:
From: Number Field in a with defining polynomial x^6 + 4*x^2 + 200
To:
Number Field in b with defining polynomial x^2 + a0 over its base field
Defn: a |--> b
sage: type(fr), type(to)
(<class 'sage.rings.number_field.morphism.RelativeNumberFieldHomomorphism_from_abs
˓→'>,
<class 'sage.rings.number_field.morphism.NumberFieldHomomorphism_im_gens'>)
sage: M.<c> = L.absolute_field(); M
Number Field in c with defining polynomial x^6 + 4*x^2 + 200
sage: fr, to = M.structure()
sage: fr
Isomorphism map:
From: Number Field in c with defining polynomial x^6 + 4*x^2 + 200
To:
Number Field in b with defining polynomial x^2 + a0 over its base field
sage: to
Isomorphism map:
From: Number Field in b with defining polynomial x^2 + a0 over its base field
To:
Number Field in c with defining polynomial x^6 + 4*x^2 + 200
sage: type(fr), type(to)
(<class 'sage.rings.number_field.maps.MapAbsoluteToRelativeNumberField'>,
<class 'sage.rings.number_field.maps.MapRelativeToAbsoluteNumberField'>)
sage: fr(M.gen()), to(fr(M.gen())) == M.gen()
(b, True)
sage: to(L.gen()), fr(to(L.gen())) == L.gen()
(c, True)
sage: (to * fr)(M.gen()) == M.gen(), (fr * to)(L.gen()) == L.gen()
(True, True)
class sage.rings.number_field.maps. MapRelativeVectorSpaceToRelativeNumberField ( V,
K)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
EXAMPLES:
sage: L.<b> = NumberField(x^4 + 3*x^2 + 1)
sage: K = L.relativize(L.subfields(2)[0][1], 'a'); K
Number Field in a with defining polynomial x^2 - b0*x + 1 over its base field
sage: V, fr, to = K.relative_vector_space()
sage: V
Vector space of dimension 2 over Number Field in b0 with defining polynomial x^2
˓→+ 1
sage: fr
Isomorphism map:
From: Vector space of dimension 2 over Number Field in b0 with defining
˓→polynomial x^2 + 1
To:
Number Field in a with defining polynomial x^2 - b0*x + 1 over its base
˓→field
sage: type(fr)
<class 'sage.rings.number_field.maps.MapRelativeVectorSpaceToRelativeNumberField'>
sage: a0 = K.gen(); b0 = K.base_field().gen()
sage: fr(to(a0 + 2*b0)), fr(V([0, 1])), fr(V([b0, 2*b0]))
(a + 2*b0, a, 2*b0*a + b0)
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sage: (fr * to)(K.gen()) == K.gen()
True
sage: (to * fr)(V([1, 2])) == V([1, 2])
True
class sage.rings.number_field.maps. MapVectorSpaceToNumberField ( V, K)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
The map to an absolute number field from its underlying Q-vector space.
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 3*x + 1)
sage: V, fr, to = K.vector_space()
sage: V
Vector space of dimension 4 over Rational Field
sage: fr
Isomorphism map:
From: Vector space of dimension 4 over Rational Field
To:
Number Field in a with defining polynomial x^4 + 3*x + 1
sage: to
Isomorphism map:
From: Number Field in a with defining polynomial x^4 + 3*x + 1
To:
Vector space of dimension 4 over Rational Field
sage: type(fr), type(to)
(<class 'sage.rings.number_field.maps.MapVectorSpaceToNumberField'>,
<class 'sage.rings.number_field.maps.MapNumberFieldToVectorSpace'>)
sage: fr.is_injective(), fr.is_surjective()
(True, True)
sage: fr.domain(), to.codomain()
(Vector space of dimension 4 over Rational Field, Vector space of dimension 4
˓→over Rational Field)
sage: to.domain(), fr.codomain()
(Number Field in a with defining polynomial x^4 + 3*x + 1, Number Field in a with
˓→defining polynomial x^4 + 3*x + 1)
sage: fr * to
Composite map:
From: Number Field in a with defining polynomial x^4 + 3*x + 1
To:
Number Field in a with defining polynomial x^4 + 3*x + 1
Defn:
Isomorphism map:
From: Number Field in a with defining polynomial x^4 + 3*x + 1
To:
Vector space of dimension 4 over Rational Field
then
Isomorphism map:
From: Vector space of dimension 4 over Rational Field
To:
Number Field in a with defining polynomial x^4 + 3*x + 1
sage: to * fr
Composite map:
From: Vector space of dimension 4 over Rational Field
To:
Vector space of dimension 4 over Rational Field
Defn:
Isomorphism map:
From: Vector space of dimension 4 over Rational Field
To:
Number Field in a with defining polynomial x^4 + 3*x + 1
then
Isomorphism map:
From: Number Field in a with defining polynomial x^4 + 3*x + 1
To:
Vector space of dimension 4 over Rational Field
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sage: to(a), to(a + 1)
((0, 1, 0, 0), (1, 1, 0, 0))
sage: fr(to(a)), fr(V([0, 1, 2, 3]))
(a, 3*a^3 + 2*a^2 + a)
class sage.rings.number_field.maps. MapVectorSpaceToRelativeNumberField ( V, L,
from_V,
from_K)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
The isomorphism to a relative number field from its underlying Q-vector space.
MapRelativeVectorSpaceToRelativeNumberField .
Compare
EXAMPLES:
sage: L.<a, b> = NumberField([x^2 + 3, x^2 + 5])
sage: V, fr, to = L.absolute_vector_space()
sage: type(fr)
<class 'sage.rings.number_field.maps.MapVectorSpaceToRelativeNumberField'>
class sage.rings.number_field.maps. NameChangeMap ( K, L)
Bases: sage.rings.number_field.maps.NumberFieldIsomorphism
A map between two isomorphic number fields with the same defining polynomial but different variable names.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 3)
sage: L.<b> = K.change_names()
sage: from_L, to_L = L.structure()
sage: from_L
Isomorphism given by variable name change map:
From: Number Field in b with defining polynomial x^2
To:
Number Field in a with defining polynomial x^2
sage: to_L
Isomorphism given by variable name change map:
From: Number Field in a with defining polynomial x^2
To:
Number Field in b with defining polynomial x^2
sage: type(from_L), type(to_L)
(<class 'sage.rings.number_field.maps.NameChangeMap'>,
˓→field.maps.NameChangeMap'>)
- 3
- 3
- 3
- 3
<class 'sage.rings.number_
class sage.rings.number_field.maps. NumberFieldIsomorphism
Bases: sage.categories.map.Map
A base class for various isomorphisms between number fields and vector spaces.
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 3*x + 1)
sage: V, fr, to = K.vector_space()
sage: isinstance(fr, sage.rings.number_field.maps.NumberFieldIsomorphism)
True
is_injective ( )
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 3*x + 1)
sage: V, fr, to = K.vector_space()
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sage: fr.is_injective()
True
is_surjective ( )
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 3*x + 1)
sage: V, fr, to = K.vector_space()
sage: fr.is_surjective()
True
2.4 Helper classes for structural embeddings and isomorphisms of
number fields
AUTHORS:
• Julian Rueth (2014-04-03): initial version
Consider the following fields 𝐿 and 𝑀 :
sage: L.<a> = QuadraticField(2)
sage: M.<a> = L.absolute_field()
Both produce the same extension of Q. However, they should not be identical because 𝑀 carries additional information:
sage: L.structure()
(Ring Coercion endomorphism of
Ring Coercion endomorphism of
sage: M.structure()
(Isomorphism given by variable
From: Number Field in a with
To:
Number Field in a with
Isomorphism given by variable
From: Number Field in a with
To:
Number Field in a with
Number Field in a with defining polynomial x^2 - 2,
Number Field in a with defining polynomial x^2 - 2)
name change map:
defining polynomial
defining polynomial
name change map:
defining polynomial
defining polynomial
x^2 - 2
x^2 - 2,
x^2 - 2
x^2 - 2)
This used to cause trouble with caching and made (absolute) number fields not unique when they should have been.
The underlying technical problem is that the morphisms returned by structure() can only be defined once the
fields in question have been created. Therefore, these morphisms cannot be part of a key which uniquely identifies a
number field.
The classes defined in this file encapsulate information about these structure morphisms which can be passed to the
factory creating number fields. This makes it possible to distinguish number fields which only differ in terms of these
structure morphisms:
sage: L is M
False
sage: N.<a> = L.absolute_field()
sage: M is N
True
class sage.rings.number_field.structure. AbsoluteFromRelative ( other)
Bases: sage.rings.number_field.structure.NumberFieldStructure
Structure for an absolute number field created from a relative number field.
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INPUT:
•other – the number field from which this field has been created.
create_structure ( field)
Return a pair of isomorphisms which go from field to other and vice versa.
class sage.rings.number_field.structure. NameChange ( other)
Bases: sage.rings.number_field.structure.NumberFieldStructure
Structure for a number field created by a change in variable name.
INPUT:
•other – the number field from which this field has been created.
create_structure ( field)
Return a pair of isomorphisms which send the generator of field to the generator of other and vice
versa.
class sage.rings.number_field.structure. NumberFieldStructure ( other)
Bases: sage.structure.unique_representation.UniqueRepresentation
Abstract base class encapsulating information about a number fields relation to other number fields.
create_structure ( field)
Return a tuple encoding structural information about field .
OUTPUT:
Typically, the output is a pair of morphisms. The first one from field to a field from which field
has been constructed and the second one its inverse. In this case, these morphisms are used as conversion
maps between the two fields.
class sage.rings.number_field.structure. RelativeFromAbsolute ( other, gen)
Bases: sage.rings.number_field.structure.NumberFieldStructure
Structure for a relative number field created from an absolute number field.
INPUT:
•other – the (absolute) number field from which this field has been created.
•gen – the generator of the intermediate field
create_structure ( field)
Return a pair of isomorphisms which go from field to other and vice versa.
INPUT:
•field – a relative number field
class sage.rings.number_field.structure. RelativeFromRelative ( other)
Bases: sage.rings.number_field.structure.NumberFieldStructure
Structure for a relative number field created from another relative number field.
INPUT:
•other – the relative number field used in the construction, see create_structure() ; there this
field will be called field_ .
create_structure ( field)
Return a pair of isomorphisms which go from field to the relative number field (called other below)
from which field has been created and vice versa.
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The isomorphism is created via the relative number field field_ which is identical to field but is
equipped with an isomorphism to an absolute field which was used in the construction of field .
INPUT:
•field – a relative number field
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CHAPTER
THREE
ORDERS, IDEALS, IDEAL CLASSES
3.1 Orders in Number Fields
AUTHORS:
• William Stein and Robert Bradshaw (2007-09): initial version
EXAMPLES:
We define an absolute order:
sage: K.<a> = NumberField(x^2 + 1); O = K.order(2*a)
sage: O.basis()
[1, 2*a]
We compute a basis for an order in a relative extension that is generated by 2 elements:
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3]); O = K.order([3*a,2*b])
sage: O.basis()
[1, 3*a - 2*b, -6*b*a + 6, 3*a]
We compute a maximal order of a degree 10 field:
sage: K.<a> = NumberField((x+1)^10 + 17)
sage: K.maximal_order()
Maximal Order in Number Field in a with defining polynomial x^10 + 10*x^9 + 45*x^8 +
˓→120*x^7 + 210*x^6 + 252*x^5 + 210*x^4 + 120*x^3 + 45*x^2 + 10*x + 18
We compute a suborder, which has index a power of 17 in the maximal order:
sage: O = K.order(17*a); O
Order in Number Field in a with defining polynomial x^10 + 10*x^9 + 45*x^8 + 120*x^7
˓→+ 210*x^6 + 252*x^5 + 210*x^4 + 120*x^3 + 45*x^2 + 10*x + 18
sage: m = O.index_in(K.maximal_order()); m
23453165165327788911665591944416226304630809183732482257
sage: factor(m)
17^45
class sage.rings.number_field.order. AbsoluteOrder ( K, module_rep, is_maximal=None,
check=True)
Bases: sage.rings.number_field.order.Order
EXAMPLES:
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sage:
sage:
sage:
sage:
sage:
sage:
Order
from sage.rings.number_field.order import *
x = polygen(QQ)
K.<a> = NumberField(x^3+2)
V, from_v, to_v = K.vector_space()
M = span([to_v(a^2), to_v(a), to_v(1)],ZZ)
O = AbsoluteOrder(K, M); O
in Number Field in a with defining polynomial x^3 + 2
sage: M = span([to_v(a^2), to_v(a), to_v(2)],ZZ)
sage: O = AbsoluteOrder(K, M); O
Traceback (most recent call last):
...
ValueError: 1 is not in the span of the module, hence not an order.
sage: loads(dumps(O)) == O
True
Quadratic elements have a special optimized type:
absolute_discriminant ( )
Return the discriminant of this order.
EXAMPLES:
sage: K.<a> = NumberField(x^8 + x^3 - 13*x + 26)
sage: O = K.maximal_order()
sage: factor(O.discriminant())
3 * 11 * 13^2 * 613 * 1575917857
sage: L = K.order(13*a^2)
sage: factor(L.discriminant())
3^3 * 5^2 * 11 * 13^60 * 613 * 733^2 * 1575917857
sage: factor(L.index_in(O))
3 * 5 * 13^29 * 733
sage: L.discriminant() / O.discriminant() == L.index_in(O)^2
True
absolute_order ( )
Return the absolute order associated to this order, which is just this order again since this is an absolute
order.
EXAMPLES:
sage:
sage:
Order
sage:
True
K.<a> = NumberField(x^3 + 2)
O1 = K.order(a); O1
in Number Field in a with defining polynomial x^3 + 2
O1.absolute_order() is O1
basis ( )
Return the basis over Z for this order.
EXAMPLES:
sage: k.<c> = NumberField(x^3 + x^2 + 1)
sage: O = k.maximal_order(); O
Maximal Order in Number Field in c with defining polynomial x^3 + x^2 + 1
sage: O.basis()
[1, c, c^2]
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The basis is an immutable sequence:
sage: type(O.basis())
<class 'sage.structure.sequence.Sequence_generic'>
The generator functionality uses the basis method:
sage: O.0
1
sage: O.1
c
sage: O.basis()
[1, c, c^2]
sage: O.ngens()
3
change_names ( names)
Return a new order isomorphic to this one in the number field with given variable names.
EXAMPLES:
sage: R = EquationOrder(x^3 + x + 1, 'alpha'); R
Order in Number Field in alpha with defining polynomial x^3 + x + 1
sage: R.basis()
[1, alpha, alpha^2]
sage: S = R.change_names('gamma'); S
Order in Number Field in gamma with defining polynomial x^3 + x + 1
sage: S.basis()
[1, gamma, gamma^2]
discriminant ( )
Return the discriminant of this order.
EXAMPLES:
sage: K.<a> = NumberField(x^8 + x^3 - 13*x + 26)
sage: O = K.maximal_order()
sage: factor(O.discriminant())
3 * 11 * 13^2 * 613 * 1575917857
sage: L = K.order(13*a^2)
sage: factor(L.discriminant())
3^3 * 5^2 * 11 * 13^60 * 613 * 733^2 * 1575917857
sage: factor(L.index_in(O))
3 * 5 * 13^29 * 733
sage: L.discriminant() / O.discriminant() == L.index_in(O)^2
True
index_in ( other)
Return the index of self in other. This is a lattice index, so it is a rational number if self isn’t contained in
other.
INPUT:
•other – another absolute order with the same ambient number field.
OUTPUT:
a rational number
EXAMPLES:
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sage:
sage:
sage:
sage:
5
k.<i> = NumberField(x^2 + 1)
O1 = k.order(i)
O5 = k.order(5*i)
O5.index_in(O1)
sage: k.<a> = NumberField(x^3 + x^2 - 2*x+8)
sage: o = k.maximal_order()
sage: o
Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x +
˓→8
sage: O1 = k.order(a); O1
Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
sage: O1.index_in(o)
2
sage: O2 = k.order(1+2*a); O2
Order in Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
sage: O1.basis()
[1, a, a^2]
sage: O2.basis()
[1, 2*a, 4*a^2]
sage: o.index_in(O2)
1/16
intersection ( other)
Return the intersection of this order with another order.
EXAMPLES:
sage: k.<i> = NumberField(x^2 + 1)
sage: O6 = k.order(6*i)
sage: O9 = k.order(9*i)
sage: O6.basis()
[1, 6*i]
sage: O9.basis()
[1, 9*i]
sage: O6.intersection(O9).basis()
[1, 18*i]
sage: (O6 & O9).basis()
[1, 18*i]
sage: (O6 + O9).basis()
[1, 3*i]
module ( )
Returns the underlying free module corresponding to this order, embedded in the vector space corresponding to the ambient number field.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + x + 3)
sage: m = k.order(3*a); m
Order in Number Field in a with defining polynomial x^3 + x + 3
sage: m.module()
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[1 0 0]
[0 3 0]
[0 0 9]
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sage.rings.number_field.order. EisensteinIntegers ( names=’omega’)
Return the ring of√
Eisenstein integers, that is all complex numbers of the form 𝑎 + 𝑏𝜔 with 𝑎 and 𝑏 integers and
𝑜𝑚𝑒𝑔𝑎 = (−1 + −3)/2.
EXAMPLES:
sage: R.<omega> = EisensteinIntegers()
sage: R
Eisenstein Integers in Number Field in omega with defining polynomial x^2 + x + 1
sage: factor(3 + omega)
(omega) * (-3*omega - 2)
sage: CC(omega)
-0.500000000000000 + 0.866025403784439*I
sage: omega.minpoly()
x^2 + x + 1
sage: EisensteinIntegers().basis()
[1, omega]
sage.rings.number_field.order. EquationOrder ( f, names, **kwds)
Return the equation order generated by a root of the irreducible polynomial f or list of polynomials 𝑓 (to construct a relative equation order).
IMPORTANT: Note that the generators of the returned order need not be roots of 𝑓 , since the generators of an
order are – in Sage – module generators.
EXAMPLES:
sage: O.<a,b> = EquationOrder([x^2+1, x^2+2])
sage: O
Relative Order in Number Field in a with defining polynomial x^2 + 1 over its
˓→base field
sage: O.0
-b*a - 1
sage: O.1
-3*a + 2*b
Of course the input polynomial must be integral:
sage: R = EquationOrder(x^3 + x + 1/3, 'alpha'); R
Traceback (most recent call last):
...
ValueError: each generator must be integral
sage: R = EquationOrder( [x^3 + x + 1, x^2 + 1/2], 'alpha'); R
Traceback (most recent call last):
...
ValueError: each generator must be integral
sage.rings.number_field.order. GaussianIntegers ( names=’I’)
Return
√ the ring of Gaussian integers, that is all complex numbers of the form 𝑎 + 𝑏𝐼 with 𝑎 and 𝑏 integers and
𝐼 = −1.
EXAMPLES:
sage: ZZI.<I> = GaussianIntegers()
sage: ZZI
Gaussian Integers in Number Field in I with defining polynomial x^2 + 1
sage: factor(3 + I)
(-I) * (I + 1) * (2*I + 1)
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sage: CC(I)
1.00000000000000*I
sage: I.minpoly()
x^2 + 1
sage: GaussianIntegers().basis()
[1, I]
class sage.rings.number_field.order. Order ( K, is_maximal)
Bases: sage.rings.ring.IntegralDomain
An order in a number field.
An order is a subring of the number field that has Z-rank equal to the degree of the number field over Q.
EXAMPLES:
sage: K.<theta> = NumberField(x^4 + x + 17)
sage: K.maximal_order()
Maximal Order in Number Field in theta with defining polynomial
sage: R = K.order(17*theta); R
Order in Number Field in theta with defining polynomial x^4 + x
sage: R.basis()
[1, 17*theta, 289*theta^2, 4913*theta^3]
sage: R = K.order(17*theta, 13*theta); R
Order in Number Field in theta with defining polynomial x^4 + x
sage: R.basis()
[1, theta, theta^2, theta^3]
sage: R = K.order([34*theta, 17*theta + 17]); R
Order in Number Field in theta with defining polynomial x^4 + x
x^4 + x + 17
+ 17
+ 17
+ 17
sage: K.<b> = NumberField(x^4 + x^2 + 2)
sage: (b^2).charpoly().factor()
(x^2 + x + 2)^2
sage: K.order(b^2)
Traceback (most recent call last):
...
ValueError: the rank of the span of gens is wrong
absolute_degree ( )
Returns the absolute degree of this order, ie the degree of this order over Z.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 2)
sage: O = K.maximal_order()
sage: O.absolute_degree()
3
ambient ( )
Return the ambient number field that contains self.
This is the same as self.number_field() and self.fraction_field()
EXAMPLES:
sage:
sage:
sage:
Order
sage:
200
k.<z> = NumberField(x^2 - 389)
o = k.order(389*z + 1)
o
in Number Field in z with defining polynomial x^2 - 389
o.basis()
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[1, 389*z]
sage: o.ambient()
Number Field in z with defining polynomial x^2 - 389
basis ( )
Return a basis over Z of this order.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + x^2 - 16*x + 16)
sage: O = K.maximal_order(); O
Maximal Order in Number Field in a with defining polynomial x^3 + x^2 - 16*x
˓→+ 16
sage: O.basis()
[1, 1/4*a^2 + 1/4*a, a^2]
class_group ( proof=None, names=’c’)
Return the class group of this order.
(Currently only implemented for the maximal order.)
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 5077)
sage: O = k.maximal_order(); O
Maximal Order in Number Field in a with defining polynomial x^2 + 5077
sage: O.class_group()
Class group of order 22 with structure C22 of Number Field in a with defining
˓→polynomial x^2 + 5077
class_number ( proof=None)
Return the class number of this order.
EXAMPLES:
sage: ZZ[2^(1/3)].class_number()
1
sage: QQ[sqrt(-23)].maximal_order().class_number()
3
Note that non-maximal orders aren’t supported yet:
sage: ZZ[3*sqrt(-3)].class_number()
Traceback (most recent call last):
...
NotImplementedError: computation of class numbers of non-maximal orders is
˓→not implemented
coordinates ( x)
Returns the coordinate vector of 𝑥 with respect to this order.
INPUT:
•x – an element of the number field of this order.
OUTPUT:
A vector of length 𝑛 (the degree of the field) giving the coordinates of 𝑥 with respect to the integral basis
of the order. In general this will be a vector of rationals; it will consist of integers if and only if 𝑥 is in the
order.
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AUTHOR: John Cremona 2008-11-15
ALGORITHM:
Uses linear algebra. The change-of-basis matrix is cached. Provides simpler implementations for
_contains_() , is_integral() and smallest_integer() .
EXAMPLES:
sage: K.<i> = QuadraticField(-1)
sage: OK = K.ring_of_integers()
sage: OK_basis = OK.basis(); OK_basis
[1, i]
sage: a = 23-14*i
sage: acoords = OK.coordinates(a); acoords
(23, -14)
sage: sum([OK_basis[j]*acoords[j] for j in range(2)]) == a
True
sage: OK.coordinates((120+340*i)/8)
(15, 85/2)
sage: O = K.order(3*i)
sage: O.is_maximal()
False
sage: O.index_in(OK)
3
sage: acoords = O.coordinates(a); acoords
(23, -14/3)
sage: sum([O.basis()[j]*acoords[j] for j in range(2)]) == a
True
degree ( )
Return the degree of this order, which is the rank of this order as a Z-module.
EXAMPLES:
sage:
sage:
sage:
3
sage:
3
k.<c> = NumberField(x^3 + x^2 - 2*x+8)
o = k.maximal_order()
o.degree()
o.rank()
fraction_field ( )
Return the fraction field of this order, which is the ambient number field.
EXAMPLES:
sage: K.<b> = NumberField(x^4 + 17*x^2 + 17)
sage: O = K.order(17*b); O
Order in Number Field in b with defining polynomial x^4 + 17*x^2 + 17
sage: O.fraction_field()
Number Field in b with defining polynomial x^4 + 17*x^2 + 17
fractional_ideal ( *args, **kwds)
Return the fractional ideal of the maximal order with given generators.
EXAMPLES:
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sage: K.<a> = NumberField(x^2 + 2)
sage: R = K.maximal_order()
sage: R.fractional_ideal(2/3 + 7*a, a)
Fractional ideal (1/3*a)
free_module ( )
Return the free Z-module contained in the vector space associated to the ambient number field, that corresponds to this ideal.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: O = K.maximal_order(); O.basis()
[1, 1/2*a^2 + 1/2*a, a^2]
sage: O.free_module()
Free module of degree 3 and rank 3 over Integer Ring
User basis matrix:
[ 1
0
0]
[ 0 1/2 1/2]
[ 0
0
1]
An example in a relative extension. Notice that the module is a Z-module in the absolute_field associated
to the relative field:
sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: O = K.maximal_order(); O.basis()
[(-3/2*b - 5)*a + 7/2*b - 2, -3*a + 2*b, -2*b*a - 3, -7*a + 5*b]
sage: O.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[1/4 1/4 3/4 3/4]
[ 0 1/2
0 1/2]
[ 0
0
1
0]
[ 0
0
0
1]
gen ( i)
Return 𝑖‘th module generator of this order.
EXAMPLES:
sage: K.<c> = NumberField(x^3 + 2*x + 17)
sage: O = K.maximal_order(); O
Maximal Order in Number Field in c with defining polynomial x^3 + 2*x + 17
sage: O.basis()
[1, c, c^2]
sage: O.gen(1)
c
sage: O.gen(2)
c^2
sage: O.gen(5)
Traceback (most recent call last):
...
IndexError: no 5th generator
sage: O.gen(-1)
Traceback (most recent call last):
...
IndexError: no -1th generator
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gens ( )
Deprecated alias for basis() .
Note: For a (much smaller) list of ring generators use ring_generators() .
EXAMPLES:
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: O = K.maximal_order()
sage: O.gens()
doctest:...: DeprecationWarning: the gens() method is deprecated, use basis()
˓→or ring_generators() instead
See http://trac.sagemath.org/15348 for details.
[1, 1/2*a^2 + 1/2*a, a^2]
ideal ( *args, **kwds)
Return the integral ideal with given generators.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 7)
sage: R = K.maximal_order()
sage: R.ideal(2/3 + 7*a, a)
Traceback (most recent call last):
...
ValueError: ideal must be integral; use fractional_ideal to create a non˓→integral ideal.
sage: R.ideal(7*a, 77 + 28*a)
Fractional ideal (7)
sage: R = K.order(4*a)
sage: R.ideal(8)
Traceback (most recent call last):
...
NotImplementedError: ideals of non-maximal orders not yet supported.
This function is called implicitly below:
sage: R = EquationOrder(x^2 + 2, 'a'); R
Order in Number Field in a with defining polynomial x^2 + 2
sage: (3,15)*R
Fractional ideal (3)
The zero ideal is handled properly:
sage: R.ideal(0)
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
integral_closure ( )
Return the integral closure of this order.
EXAMPLES:
sage: K.<a> = QuadraticField(5)
sage: O2 = K.order(2*a); O2
Order in Number Field in a with defining polynomial x^2 - 5
sage: O2.integral_closure()
Maximal Order in Number Field in a with defining polynomial x^2 - 5
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sage: OK = K.maximal_order()
sage: OK is OK.integral_closure()
True
is_field ( proof=True)
Return False (because an order is never a field).
EXAMPLES:
sage: L.<alpha> = NumberField(x**4 - x**2 + 7)
sage: O = L.maximal_order() ; O.is_field()
False
sage: CyclotomicField(12).ring_of_integers().is_field()
False
is_integrally_closed ( )
Return True if this ring is integrally closed, i.e., is equal to the maximal order.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 189*x + 394)
sage: R = K.order(2*a)
sage: R.is_integrally_closed()
False
sage: R
Order in Number Field in a with defining polynomial x^2 + 189*x + 394
sage: S = K.maximal_order(); S
Maximal Order in Number Field in a with defining polynomial x^2 + 189*x + 394
sage: S.is_integrally_closed()
True
is_maximal ( )
Returns True if this is the maximal order.
EXAMPLES:
sage: k.<i> = NumberField(x^2 + 1)
sage: O3 = k.order(3*i); O5 = k.order(5*i); Ok = k.maximal_order(); Osum =
˓→O3 + O5
sage: Osum.is_maximal()
True
sage: O3.is_maximal()
False
sage: O5.is_maximal()
False
sage: Ok.is_maximal()
True
An example involving a relative order::
sage: K.<a,b> = NumberField([x^2 + 1, x^2 - 3]); O = K.order([3*a,2*b]); O
Relative Order in Number Field in a with defining polynomial x^2 + 1 over
˓→its base field
sage: O.is_maximal()
False
is_noetherian ( )
Return True (because orders are always Noetherian)
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EXAMPLES:
sage:
sage:
True
sage:
sage:
True
L.<alpha> = NumberField(x**4 - x**2 + 7)
O = L.maximal_order() ; O.is_noetherian()
E.<w> = NumberField(x^2 - x + 2)
OE = E.ring_of_integers(); OE.is_noetherian()
is_suborder ( other)
Return True if self and other are both orders in the same ambient number field and self is a subset of other.
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
True
sage:
False
sage:
False
W.<i> = NumberField(x^2 + 1)
O5 = W.order(5*i)
O10 = W.order(10*i)
O15 = W.order(15*i)
O15.is_suborder(O5)
O5.is_suborder(O15)
O10.is_suborder(O15)
We create another isomorphic but different field:
sage: W2.<j> = NumberField(x^2 + 1)
sage: P5 = W2.order(5*j)
This is False because the ambient number fields are not equal.:
sage: O5.is_suborder(P5)
False
We create a field that contains (in no natural way!) W, and of course again is_suborder returns False:
sage: K.<z> = NumberField(x^4 + 1)
sage: M = K.order(5*z)
sage: O5.is_suborder(M)
False
krull_dimension ( )
Return the Krull dimension of this order, which is 1.
EXAMPLES:
sage:
sage:
sage:
1
sage:
sage:
1
K.<a> = QuadraticField(5)
OK = K.maximal_order()
OK.krull_dimension()
O2 = K.order(2*a)
O2.krull_dimension()
ngens ( )
Return the number of module generators of this order.
EXAMPLES:
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sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: O = K.maximal_order()
sage: O.ngens()
3
number_field ( )
Return the number field of this order, which is the ambient number field that this order is embedded in.
EXAMPLES:
sage: K.<b> = NumberField(x^4 + x^2 + 2)
sage: O = K.order(2*b); O
Order in Number Field in b with defining polynomial x^4 + x^2 + 2
sage: O.basis()
[1, 2*b, 4*b^2, 8*b^3]
sage: O.number_field()
Number Field in b with defining polynomial x^4 + x^2 + 2
sage: O.number_field() is K
True
random_element ( *args, **kwds)
Return a random element of this order.
INPUT:
•args , kwds – parameters passed to the random integer function. See the documentation for
ZZ.random_element() for details.
OUTPUT:
A random element of this order, computed as a random Z-linear combination of the basis.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 2)
sage: OK = K.ring_of_integers()
sage: OK.random_element() # random output
-2*a^2 - a - 2
sage: OK.random_element(distribution="uniform") # random output
-a^2 - 1
sage: OK.random_element(-10,10) # random output
-10*a^2 - 9*a - 2
sage: K.order(a).random_element() # random output
a^2 - a - 3
sage: K.<z> = CyclotomicField(17)
sage: OK = K.ring_of_integers()
sage: OK.random_element() # random output
z^15 - z^11 - z^10 - 4*z^9 + z^8 + 2*z^7 + z^6 - 2*z^5 - z^4 - 445*z^3 - 2*z^
˓→2 - 15*z - 2
sage: OK.random_element().is_integral()
True
sage: OK.random_element().parent() is OK
True
A relative example:
sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
sage: OK = K.ring_of_integers()
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sage: OK.random_element() # random output
(42221/2*b + 61/2)*a + 7037384*b + 7041
sage: OK.random_element().is_integral() # random output
True
sage: OK.random_element().parent() is OK # random output
True
An example in a non-maximal order:
sage: K.<a> = QuadraticField(-3)
sage: R = K.ring_of_integers()
sage: A = K.order(a)
sage: A.index_in(R)
2
sage: R.random_element() # random output
-39/2*a - 1/2
sage: A.random_element() # random output
2*a - 1
sage: A.random_element().is_integral()
True
sage: A.random_element().parent() is A
True
rank ( )
Return the rank of this order, which is the rank of the underlying Z-module, or the degree of the ambient
number field that contains this order.
This is a synonym for degree() .
EXAMPLES:
sage: k.<c> = NumberField(x^5 + x^2 + 1)
sage: o = k.maximal_order(); o
Maximal Order in Number Field in c with defining polynomial x^5 + x^2 + 1
sage: o.rank()
5
residue_field ( prime, names=None, check=False)
Return the residue field of this order at a given prime, ie 𝑂/𝑝𝑂.
INPUT:
•prime – a prime ideal of the maximal order in this number field.
•names – the name of the variable in the residue field
•check – whether or not to check the primality of prime.
OUTPUT:
The residue field at this prime.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^4+3*x^2-17)
sage: P = K.ideal(61).factor()[0][0]
sage: OK = K.maximal_order()
sage: OK.residue_field(P)
Residue field in abar of Fractional ideal (61, a^2 + 30)
sage: Fp.<b> = OK.residue_field(P)
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sage: Fp
Residue field in b of Fractional ideal (61, a^2 + 30)
ring_generators ( )
Return generators for self as a ring.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: O = K.maximal_order(); O
Gaussian Integers in Number Field in i with defining polynomial x^2 + 1
sage: O.ring_generators()
[i]
This is an example where 2 generators are required (because 2 is an essential discriminant divisor).:
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: O = K.maximal_order(); O.basis()
[1, 1/2*a^2 + 1/2*a, a^2]
sage: O.ring_generators()
[1/2*a^2 + 1/2*a, a^2]
An example in a relative number field:
sage: K.<a, b> = NumberField([x^2 + x + 1, x^3 - 3])
sage: O = K.maximal_order()
sage: O.ring_generators()
[(-5/3*b^2 + 3*b - 2)*a - 7/3*b^2 + b + 3, (-5*b^2 - 9)*a - 5*b^2 - b, (-6*b^
˓→2 - 11)*a - 6*b^2 - b]
zeta ( n=2, all=False)
Return a primitive n-th root of unity in this order, if it contains one. If all is True, return all of them.
EXAMPLES:
sage: F.<alpha> = NumberField(x**2+3)
sage: F.ring_of_integers().zeta(6)
1/2*alpha + 1/2
sage: O = F.order([3*alpha])
sage: O.zeta(3)
Traceback (most recent call last):
...
ArithmeticError: There are no 3rd roots of unity in self.
class sage.rings.number_field.order. RelativeOrder ( K,
absolute_order,
is_maximal=None, check=True)
Bases: sage.rings.number_field.order.Order
A relative order in a number field.
A relative order is an order in some relative number field
Invariants of this order may be computed with respect to the contained order.
absolute_discriminant ( )
Return the absolute discriminant of self, which is the discriminant of the absolute order associated to self.
OUTPUT:
an integer
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EXAMPLES:
sage: R = EquationOrder([x^2 + 1, x^3 + 2], 'a,b')
sage: d = R.absolute_discriminant(); d
-746496
sage: d is R.absolute_discriminant()
True
sage: factor(d)
-1 * 2^10 * 3^6
absolute_order ( names=’z’)
Return underlying absolute order associated to this relative order.
INPUT:
•names – string (default: ‘z’); name of generator of absolute extension.
Note: There is a default variable name, since this absolute order is frequently used for internal algorithms.
EXAMPLES:
sage: R = EquationOrder([x^2 + 1, x^2 - 5], 'i,g'); R
Relative Order in Number Field in i with defining polynomial x^2 + 1 over its
˓→base field
sage: R.basis()
[1, 6*i - g, -g*i + 2, 7*i - g]
sage: S = R.absolute_order(); S
Order in Number Field in z with defining polynomial x^4 - 8*x^2 + 36
sage: S.basis()
[1, 5/12*z^3 + 1/6*z, 1/2*z^2, 1/2*z^3]
We compute a relative order in alpha0, alpha1, then make the number field that contains the absolute order
be called gamma.:
sage: R = EquationOrder( [x^2 + 2, x^2 - 3], 'alpha'); R
Relative Order in Number Field in alpha0 with defining polynomial x^2 + 2
˓→over its base field
sage: R.absolute_order('gamma')
Order in Number Field in gamma with defining polynomial x^4 - 2*x^2 + 25
sage: R.absolute_order('gamma').basis()
[1/2*gamma^2 + 1/2, 7/10*gamma^3 + 1/10*gamma, gamma^2, gamma^3]
basis ( )
Return a basis for this order as Z-module.
EXAMPLES:
sage: K.<a,b> = NumberField([x^2+1, x^2+3])
sage: O = K.order([a,b])
sage: O.basis()
[1, -2*a + b, -b*a - 2, -5*a + 3*b]
sage: z = O.1; z
-2*a + b
sage: z.absolute_minpoly()
x^4 + 14*x^2 + 1
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index_in ( other)
Return the index of self in other. This is a lattice index, so it is a rational number if self isn’t contained in
other.
INPUT:
•other – another order with the same ambient absolute number field.
OUTPUT:
a rational number
EXAMPLES:
sage:
sage:
sage:
sage:
729/8
sage:
8/729
K.<a,b> = NumberField([x^3 + x + 3, x^2 + 1])
R1 = K.order([3*a, 2*b])
R2 = K.order([a, 4*b])
R1.index_in(R2)
R2.index_in(R1)
is_suborder ( other)
Returns true if self is a subset of the order other.
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
False
sage:
True
sage:
True
sage:
True
sage:
True
K.<a,b> = NumberField([x^2 + 1, x^3 + 2])
R1 = K.order([a,b])
R2 = K.order([2*a,b])
R3 = K.order([a + b, b + 2*a])
R1.is_suborder(R2)
R2.is_suborder(R1)
R3.is_suborder(R1)
R1.is_suborder(R3)
R1 == R3
sage.rings.number_field.order. absolute_order_from_module_generators ( gens,
check_integral=True,
check_rank=True,
check_is_ring=True,
is_maximal=None,
allow_subfield=False)
INPUT:
•gens – list of elements of an absolute number field that generates an order in that number field as a ZZ
module.
•check_integral – check that each gen is integral
•check_rank – check that the gens span a module of the correct rank
•check_is_ring – check that the module is closed under multiplication (this is very expensive)
•is_maximal – bool (or None); set if maximality of the generated order is known
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OUTPUT:
an absolute order
EXAMPLES: We have to explicitly import the function, since it isn’t meant for regular usage:
sage: from sage.rings.number_field.order import absolute_order_from_module_
˓→generators
sage: K.<a> = NumberField(x^4 - 5)
sage: O = K.maximal_order(); O
Maximal Order in Number Field in a with defining polynomial x^4 - 5
sage: O.basis()
[1/2*a^2 + 1/2, 1/2*a^3 + 1/2*a, a^2, a^3]
sage: O.module()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/2
0 1/2
0]
[ 0 1/2
0 1/2]
[ 0
0
1
0]
[ 0
0
0
1]
sage: g = O.basis(); g
[1/2*a^2 + 1/2, 1/2*a^3 + 1/2*a, a^2, a^3]
sage: absolute_order_from_module_generators(g)
Order in Number Field in a with defining polynomial x^4 - 5
We illustrate each check flag – the output is the same but in case the function would run ever so slightly faster:
sage:
Order
sage:
Order
sage:
Order
absolute_order_from_module_generators(g, check_is_ring=False)
in Number Field in a with defining polynomial x^4 - 5
absolute_order_from_module_generators(g, check_rank=False)
in Number Field in a with defining polynomial x^4 - 5
absolute_order_from_module_generators(g, check_integral=False)
in Number Field in a with defining polynomial x^4 - 5
Next we illustrate constructing “fake” orders to illustrate turning off various check flags:
sage: k.<i> = NumberField(x^2 + 1)
sage: R = absolute_order_from_module_generators([2, 2*i], check_is_ring=False); R
Order in Number Field in i with defining polynomial x^2 + 1
sage: R.basis()
[2, 2*i]
sage: R = absolute_order_from_module_generators([k(1)], check_rank=False); R
Order in Number Field in i with defining polynomial x^2 + 1
sage: R.basis()
[1]
If the order contains a non-integral element, even if we don’t check that, we’ll find that the rank is wrong or that
the order isn’t closed under multiplication:
sage: absolute_order_from_module_generators([1/2, i], check_integral=False)
Traceback (most recent call last):
...
ValueError: the module span of the gens is not closed under multiplication.
sage: R = absolute_order_from_module_generators([1/2, i], check_is_ring=False,
˓→check_integral=False); R
Order in Number Field in i with defining polynomial x^2 + 1
sage: R.basis()
[1/2, i]
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We turn off all check flags and make a really messed up order:
sage: R = absolute_order_from_module_generators([1/2, i], check_is_ring=False,
˓→check_integral=False, check_rank=False); R
Order in Number Field in i with defining polynomial x^2 + 1
sage: R.basis()
[1/2, i]
An order that lives in a subfield:
sage: F.<alpha> = NumberField(x**4+3)
sage: F.order([alpha**2], allow_subfield=True)
Order in Number Field in alpha with defining polynomial x^4 + 3
sage.rings.number_field.order. absolute_order_from_ring_generators ( gens,
check_is_integral=True,
check_rank=True,
is_maximal=None,
allow_subfield=False)
INPUT:
•gens – list of integral elements of an absolute order.
•check_is_integral – bool (default: True), whether to check that each generator is integral.
•check_rank – bool (default: True), whether to check that the ring generated by gens is of full rank.
•is_maximal – bool (or None); set if maximality of the generated order is known
•allow_subfield – bool (default: False), if True and the generators do not generate an order, i.e., they
generate a subring of smaller rank, instead of raising an error, return an order in a smaller number field.
EXAMPLES:
sage: K.<a> = NumberField(x^4 - 5)
sage: K.order(a)
Order in Number Field in a with defining polynomial x^4 - 5
We have to explicitly import this function, since typically it is called with K.order as above.:
sage: from sage.rings.number_field.order import absolute_order_from_ring_
˓→generators
sage: absolute_order_from_ring_generators([a])
Order in Number Field in a with defining polynomial x^4 - 5
sage: absolute_order_from_ring_generators([3*a, 2, 6*a+1])
Order in Number Field in a with defining polynomial x^4 - 5
If one of the inputs is non-integral, it is an error.:
sage: absolute_order_from_ring_generators([a/2])
Traceback (most recent call last):
...
ValueError: each generator must be integral
If the gens do not generate an order, i.e., generate a ring of full rank, then it is an error.:
sage: absolute_order_from_ring_generators([a^2])
Traceback (most recent call last):
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...
ValueError: the rank of the span of gens is wrong
Both checking for integrality and checking for full rank can be turned off in order to save time, though one can
get nonsense as illustrated below.:
sage:
Order
sage:
Order
absolute_order_from_ring_generators([a/2], check_is_integral=False)
in Number Field in a with defining polynomial x^4 - 5
absolute_order_from_ring_generators([a^2], check_rank=False)
in Number Field in a with defining polynomial x^4 - 5
sage.rings.number_field.order. each_is_integral ( v)
Return True if each element of the list v of elements of a number field is integral.
EXAMPLES:
sage:
sage:
sage:
True
sage:
False
W.<sqrt5> = NumberField(x^2 - 5)
from sage.rings.number_field.order import each_is_integral
each_is_integral([sqrt5, 2, (1+sqrt5)/2])
each_is_integral([sqrt5, (1+sqrt5)/3])
sage.rings.number_field.order. is_NumberFieldOrder ( R)
Return True if R is either an order in a number field or is the ring Z of integers.
EXAMPLES:
sage:
sage:
True
sage:
True
sage:
False
sage:
False
from sage.rings.number_field.order import is_NumberFieldOrder
is_NumberFieldOrder(NumberField(x^2+1,'a').maximal_order())
is_NumberFieldOrder(ZZ)
is_NumberFieldOrder(QQ)
is_NumberFieldOrder(45)
sage.rings.number_field.order. relative_order_from_ring_generators ( gens,
check_is_integral=True,
check_rank=True,
is_maximal=None,
allow_subfield=False)
INPUT:
•gens – list of integral elements of an absolute order.
•check_is_integral – bool (default: True), whether to check that each generator is integral.
•check_rank – bool (default: True), whether to check that the ring generated by gens is of full rank.
•is_maximal – bool (or None); set if maximality of the generated order is known
EXAMPLES: We have to explicitly import this function, since it isn’t meant for regular usage:
sage: from sage.rings.number_field.order import relative_order_from_ring_
˓→generators
sage: K.<i, a> = NumberField([x^2 + 1, x^2 - 17])
sage: R = K.base_field().maximal_order()
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sage: S = relative_order_from_ring_generators([i,a]); S
Relative Order in Number Field in i with defining polynomial x^2 + 1 over its
˓→base field
Basis for the relative order, which is obtained by computing the algebra generated by i and a:
sage: S.basis()
[1, 7*i - 2*a, -a*i + 8, 25*i - 7*a]
3.2 Number Field Ideals
AUTHORS:
• Steven Sivek (2005-05-16)
• William Stein (2007-09-06): vastly improved the doctesting
• William Stein and John Cremona (2007-01-28): new class NumberFieldFractionalIdeal now used for all except
the 0 ideal
• Radoslav Kirov and Alyson Deines (2010-06-22): prime_to_S_part, is_S_unit, is_S_integral
We test that pickling works:
sage: K.<a> = NumberField(x^2 - 5)
sage: I = K.ideal(2/(5+a))
sage: I == loads(dumps(I))
True
class sage.rings.number_field.number_field_ideal. LiftMap ( OK, M_OK_map, Q, I)
Class to hold data needed by lifting maps from residue fields to number field orders.
class sage.rings.number_field.number_field_ideal. NumberFieldFractionalIdeal ( field,
gens,
coerce=True)
Bases:
sage.structure.element.MultiplicativeGroupElement
,
sage.rings.number_field.number_field_ideal.NumberFieldIdeal
A fractional ideal in a number field.
denominator ( )
Return the denominator ideal of this fractional ideal. Each fractional ideal has a unique expression as 𝑁/𝐷
where 𝑁 , 𝐷 are coprime integral ideals; the denominator is 𝐷.
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
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sage: I.numerator()/I.denominator() == I
True
divides ( other)
Returns True if this ideal divides other and False otherwise.
EXAMPLES:
sage: K.<a> = CyclotomicField(11); K
Cyclotomic Field of order 11 and degree 10
sage: I = K.factor(31)[0][0]; I
Fractional ideal (31, a^5 + 10*a^4 - a^3 + a^2 + 9*a - 1)
sage: I.divides(I)
True
sage: I.divides(31)
True
sage: I.divides(29)
False
element_1_mod ( other)
Returns an element 𝑟 in this ideal such that 1 − 𝑟 is in other
An error is raised if either ideal is not integral of if they are not coprime.
INPUT:
•other – another ideal of the same field, or generators of an ideal.
OUTPUT:
An element 𝑟 of the ideal self such that 1 − 𝑟 is in the ideal other
AUTHOR: Maite Aranes (modified to use PARI’s pari:idealaddtoone by Francis Clarke)
EXAMPLES:
sage: K.<a> = NumberField(x^3-2)
sage: A = K.ideal(a+1); A; A.norm()
Fractional ideal (a + 1)
3
sage: B = K.ideal(a^2-4*a+2); B; B.norm()
Fractional ideal (a^2 - 4*a + 2)
68
sage: r = A.element_1_mod(B); r
-a^2 + 4*a - 1
sage: r in A
True
sage: 1-r in B
True
euler_phi ( )
Returns the Euler 𝜙-function of this integral ideal.
This is the order of the multiplicative group of the quotient modulo the ideal.
An error is raised if the ideal is not integral.
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal(2+i)
sage: [r for r in I.residues() if I.is_coprime(r)]
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[-2*i, -i, i, 2*i]
sage: I.euler_phi()
4
sage: J = I^3
sage: J.euler_phi()
100
sage: len([r for r in J.residues() if J.is_coprime(r)])
100
sage: J = K.ideal(3-2*i)
sage: I.is_coprime(J)
True
sage: I.euler_phi()*J.euler_phi() == (I*J).euler_phi()
True
sage: L.<b> = K.extension(x^2 - 7)
sage: L.ideal(3).euler_phi()
64
factor ( )
Factorization of this ideal in terms of prime ideals.
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: F = I.factor(); F
(Fractional ideal (19, 1/2*a^2 + a - 17/2)) * (Fractional ideal (19, 1/2*a^2 ˓→ a - 17/2))
sage: type(F)
<class 'sage.structure.factorization.Factorization'>
sage: list(F)
[(Fractional ideal (19, 1/2*a^2 + a - 17/2), 1), (Fractional ideal (19, 1/2*a^
˓→2 - a - 17/2), 1)]
sage: F.prod()
Fractional ideal (19)
idealcoprime ( J)
Returns l such that l*self is coprime to J.
INPUT:
•J - another integral ideal of the same field as self, which must also be integral.
OUTPUT:
•l - an element such that l*self is coprime to the ideal J
TODO: Extend the implementation to non-integral ideals.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23)
sage: A = k.ideal(a+1)
sage: B = k.ideal(3)
sage: A.is_coprime(B)
False
sage: lam = A.idealcoprime(B); lam
-1/6*a + 1/6
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sage: (lam*A).is_coprime(B)
True
ALGORITHM: Uses Pari function pari:idealcoprime.
ideallog ( x, gens=None, check=True)
Returns the discrete logarithm of x with respect to the generators given in the bid structure of the ideal
self, or with respect to the generators gens if these are given.
INPUT:
•x - a non-zero element of the number field of self, which must have valuation equal to 0 at all prime
ideals in the support of the ideal self.
•gens - a list of elements of the number field which generate (𝑅/𝐼)* , where 𝑅 is the ring of integers
of the field and 𝐼 is this ideal, or None . If None , use the generators calculated by idealstar() .
•check - if True, do a consistency check on the results. Ignored if gens is None.
OUTPUT:
∏︀
•l - a list of non-negative integers (𝑥𝑖 ) such that 𝑥 = 𝑖 𝑔𝑖𝑥𝑖 in (𝑅/𝐼)* , where 𝑥𝑖 are the generators, and the list (𝑥𝑖 ) is lexicographically minimal with respect to this requirement. If the 𝑥𝑖
generate independent cyclic factors of order 𝑑𝑖 , as is the case for the default generators calculated by
idealstar() , this just means that 0 ≤ 𝑥𝑖 < 𝑑𝑖 .
A ValueError will be raised if the elements specified in gens do not in fact generate the unit group
(even if the element 𝑥 is in the subgroup they generate).
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
sage:
True
sage:
a^2 -
k.<a> = NumberField(x^3 - 11)
A = k.ideal(5)
G = A.idealstar(2)
l = A.ideallog(a^2 +3)
r = G(l).value()
(a^2 + 3) - r in A
A.small_residue(r) # random
2
Examples with custom generators:
sage: K.<a> = NumberField(x^2 - 7)
sage: I = K.ideal(17)
sage: I.ideallog(a + 7, [1+a, 2])
[10, 3]
sage: I.ideallog(a + 7, [2, 1+a])
[0, 118]
sage: L.<b> = NumberField(x^4 - x^3 - 7*x^2 + 3*x + 2)
sage: J = L.ideal(-b^3 - b^2 - 2)
sage: u = -14*b^3 + 21*b^2 + b - 1
sage: v = 4*b^2 + 2*b - 1
sage: J.ideallog(5+2*b, [u, v], check=True)
[4, 13]
A non-example:
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sage: I.ideallog(a + 7, [2])
Traceback (most recent call last):
...
ValueError: Given elements do not generate unit group -- they generate a
˓→subgroup of index 36
ALGORITHM: Uses Pari function pari:ideallog, and (if gens is not None) a Hermite normal form calculation to express the result in terms of the generators gens .
idealstar ( flag=1)
Returns the finite abelian group (𝑂𝐾 /𝐼)* , where I is the ideal self of the number field K, and 𝑂𝐾 is the
ring of integers of K.
INPUT:
•flag (int default 1) – when flag =2, it also computes the generators of the group (𝑂𝐾 /𝐼)* , which
takes more time. By default flag =1 (no generators are computed). In both cases the special pari
structure bid is computed as well. If flag =0 (deprecated) it computes only the group structure of
(𝑂𝐾 /𝐼)* (with generators) and not the special bid structure.
OUTPUT:
The finite abelian group (𝑂𝐾 /𝐼)* .
Note: Uses the pari function pari:idealstar. The pari function outputs a special bid structure which is
stored in the internal field _bid of the ideal (when flag=1,2). The special structure bid is used in the
pari function pari:ideallog to compute discrete logarithms.
EXAMPLES:
sage: k.<a> = NumberField(x^3 - 11)
sage: A = k.ideal(5)
sage: G = A.idealstar(); G
Multiplicative Abelian group isomorphic to C24 x C4
sage: G.gens()
(f0, f1)
sage: G = A.idealstar(2)
sage: G.gens()
(f0, f1)
sage: G.gens_values()
# random output
(2*a^2 - 1, 2*a^2 + 2*a - 2)
sage: all([G.gen(i).value() in k for i in range(G.ngens())])
True
ALGORITHM: Uses Pari function pari:idealstar
invertible_residues ( reduce=True)
Returns a iterator through a list of invertible residues modulo this integral ideal.
An error is raised if this fractional ideal is not integral.
INPUT:
•reduce - bool. If True (default), use small_residue to get small representatives of the residues.
OUTPUT:
•An iterator through a list of invertible residues modulo this ideal 𝐼, i.e. a list of elements in the ring
of integers 𝑅 representing the elements of (𝑅/𝐼)* .
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ALGORITHM: Use pari’s idealstar to find the group structure and generators of the multiplicative
group modulo the ideal.
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: ires = K.ideal(2).invertible_residues(); ires
xmrange_iter([[0, 1]], <function <lambda> at 0x...>)
sage: list(ires)
[1, -i]
sage: list(K.ideal(2+i).invertible_residues())
[1, 2, 4, 3]
sage: list(K.ideal(i).residues())
[0]
sage: list(K.ideal(i).invertible_residues())
[1]
sage: I = K.ideal(3+6*i)
sage: units=I.invertible_residues()
sage: len(list(units))==I.euler_phi()
True
sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.invertible_residues())) == I.euler_phi()
True
sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues()))
80
AUTHOR: John Cremona
invertible_residues_mod ( subgp_gens=[], reduce=True)
Returns a iterator through a list of representatives for the invertible residues modulo this integral ideal,
modulo the subgroup generated by the elements in the list subgp_gens .
INPUT:
•subgp_gens - either None or a list of elements of the number field of self. These need not be
integral, but should be coprime to the ideal self. If the list is empty or None, the function returns an
iterator through a list of representatives for the invertible residues modulo the integral ideal self.
•reduce - bool. If True (default), use small_residues to get small representatives of the
residues.
Note: See also invertible_residues() for a simpler version without the subgroup.
OUTPUT:
•An iterator through a list of representatives for the invertible residues modulo self and modulo the
group generated by subgp_gens , i.e. a list of elements in the ring of integers 𝑅 representing
the elements of (𝑅/𝐼)* /𝑈 , where 𝐼 is this ideal and 𝑈 is the subgroup of (𝑅/𝐼)* generated by
subgp_gens .
EXAMPLES:
sage: k.<a> = NumberField(x^2 +23)
sage: I = k.ideal(a)
sage: list(I.invertible_residues_mod([-1]))
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[1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9]
sage: list(I.invertible_residues_mod([1/2]))
[1, 5]
sage: list(I.invertible_residues_mod([23]))
Traceback (most recent call last):
...
TypeError: the element must be invertible mod the ideal
sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.invertible_residues_mod([]))) == I.euler_phi()
True
sage: I = K.ideal(1)
sage: list(I.invertible_residues_mod([]))
[1]
sage: K.<z> = CyclotomicField(10)
sage: len(list(K.primes_above(3)[0].invertible_residues_mod([])))
80
AUTHOR: Maite Aranes.
is_S_integral ( S)
Return True if this fractional ideal is integral with respect to the list of primes S .
INPUT:
•𝑆 - a list of prime ideals (not checked if they are indeed prime).
Note: This function assumes that 𝑆 is a list of prime ideals, but does not check this. This function will
fail if 𝑆 is not a list of prime ideals.
OUTPUT:
True, if the ideal is 𝑆-integral: that is, if the valuations of the ideal at all primes not in 𝑆 are non-negative.
False, otherwise.
EXAMPLES:
sage:
sage:
sage:
sage:
False
K.<a> = NumberField(x^2+23)
I = K.ideal(1/2)
P = K.ideal(2,1/2*a - 1/2)
I.is_S_integral([P])
sage: J = K.ideal(1/5)
sage: J.is_S_integral([K.ideal(5)])
True
is_S_unit ( S)
Return True if this fractional ideal is a unit with respect to the list of primes S .
INPUT:
•𝑆 - a list of prime ideals (not checked if they are indeed prime).
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Note: This function assumes that 𝑆 is a list of prime ideals, but does not check this. This function will
fail if 𝑆 is not a list of prime ideals.
OUTPUT:
True, if the ideal is an 𝑆-unit: that is, if the valuations of the ideal at all primes not in 𝑆 are zero. False,
otherwise.
EXAMPLES:
sage:
sage:
sage:
sage:
False
K.<a> = NumberField(x^2+23)
I = K.ideal(2)
P = I.factor()[0][0]
I.is_S_unit([P])
is_coprime ( other)
Returns True if this ideal is coprime to the other, else False.
INPUT:
•other – another ideal of the same field, or generators of an ideal.
OUTPUT:
True if self and other are coprime, else False.
Note: This function works for fractional ideals as well as integral ideals.
AUTHOR: John Cremona
EXAMPLES:
sage:
sage:
sage:
sage:
True
sage:
True
sage:
False
sage:
True
K.<i>=NumberField(x^2+1)
I = K.ideal(2+i)
J = K.ideal(2-i)
I.is_coprime(J)
(I^-1).is_coprime(J^3)
I.is_coprime(5)
I.is_coprime(6+i)
See trac ticket #4536:
sage:
sage:
sage:
sage:
False
sage:
False
E.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
OE = E.ring_of_integers()
i,j,k = [u[0] for u in factor(3*OE)]
(i/j).is_coprime(j/k)
(j/k).is_coprime(j/k)
sage: F.<a, b> = NumberField([x^2 - 2, x^2 - 3])
sage: F.ideal(3 - a*b).is_coprime(F.ideal(3))
False
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is_maximal ( )
Return True if this ideal is maximal. This is equivalent to self being prime, since it is nonzero.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True
is_trivial ( proof=None)
Returns True if this is a trivial ideal.
EXAMPLES:
sage:
sage:
sage:
False
sage:
sage:
False
sage:
True
F.<a> = QuadraticField(-5)
I = F.ideal(3)
I.is_trivial()
J = F.ideal(5)
J.is_trivial()
(I+J).is_trivial()
numerator ( )
Return the numerator ideal of this fractional ideal.
Each fractional ideal has a unique expression as 𝑁/𝐷 where 𝑁 , 𝐷 are coprime integral ideals. The
numerator is 𝑁 .
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: I = K.ideal((3+4*i)/5); I
Fractional ideal (4/5*i + 3/5)
sage: I.denominator()
Fractional ideal (2*i + 1)
sage: I.numerator()
Fractional ideal (-i - 2)
sage: I.numerator().is_integral() and I.denominator().is_integral()
True
sage: I.numerator() + I.denominator() == K.unit_ideal()
True
sage: I.numerator()/I.denominator() == I
True
prime_factors ( )
Return a list of the prime ideal factors of self
OUTPUT: list – list of prime ideals (a new list is returned each time this function is called)
EXAMPLES:
sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w+1)
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sage: I.prime_factors()
[Fractional ideal (2, 1/2*w - 1/2), Fractional ideal (2, 1/2*w + 1/2),
˓→Fractional ideal (3, 1/2*w + 1/2)]
prime_to_S_part ( S)
Return the part of this fractional ideal which is coprime to the prime ideals in the list S .
Note: This function assumes that 𝑆 is a list of prime ideals, but does not check this. This function will
fail if 𝑆 is not a list of prime ideals.
INPUT:
•𝑆 – a list of prime ideals
OUTPUT:
A fractional ideal coprime to the primes in 𝑆, whose prime factorization is that of self with the primes
in 𝑆 removed.
EXAMPLES:
sage: K.<a> = NumberField(x^2-23)
sage: I = K.ideal(24)
sage: S = [K.ideal(-a+5),K.ideal(5)]
sage: I.prime_to_S_part(S)
Fractional ideal (3)
sage: J = K.ideal(15)
sage: J.prime_to_S_part(S)
Fractional ideal (3)
sage: K.<a> = NumberField(x^5-23)
sage: I = K.ideal(24)
sage: S = [K.ideal(15161*a^4 + 28383*a^3 + 53135*a^2 + 99478*a + 186250),K.
˓→ideal(2*a^4 + 3*a^3 + 4*a^2 + 15*a + 11), K.ideal(101)]
sage: I.prime_to_S_part(S)
Fractional ideal (24)
prime_to_idealM_part ( M)
Version for integral ideals of the prime_to_m_part function over Z. Returns the largest divisor of
self that is coprime to the ideal M .
INPUT:
•M – an integral ideal of the same field, or generators of an ideal
OUTPUT:
An ideal which is the largest divisor of self that is coprime to 𝑀 .
AUTHOR: Maite Aranes
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(a+1)
sage: M = k.ideal(2, 1/2*a - 1/2)
sage: J = I.prime_to_idealM_part(M); J
Fractional ideal (12, 1/2*a + 13/2)
sage: J.is_coprime(M)
True
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sage: J = I.prime_to_idealM_part(2); J
Fractional ideal (3, 1/2*a + 1/2)
sage: J.is_coprime(M)
True
ramification_index ( )
Return the ramification index of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.
The ramification index is the power of this prime appearing in the factorization of the prime in Z that this
prime lies over.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: f = K.factor(2); f
(Fractional ideal (a))^2
sage: f[0][0].ramification_index()
2
sage: K.ideal(13).ramification_index()
1
sage: K.ideal(17).ramification_index()
Traceback (most recent call last):
...
ValueError: Fractional ideal (17) is not a prime ideal
ray_class_number ( )
Return the order of the ray class group modulo this ideal. This is a wrapper around Pari’s pari:bnrclassno
function.
EXAMPLES:
sage: K.<z> = QuadraticField(-23)
sage: p = K.primes_above(3)[0]
sage: p.ray_class_number()
3
sage:
sage:
sage:
sage:
5184
x = polygen(K)
L.<w> = K.extension(x^3 - z)
I = L.ideal(5)
I.ray_class_number()
reduce ( f )
Return the canonical reduction of the element of 𝑓 modulo the ideal 𝐼 (=self). This is an element of 𝑅 (the
ring of integers of the number field) that is equivalent modulo 𝐼 to 𝑓 .
An error is raised if this fractional ideal is not integral or the element 𝑓 is not integral.
INPUT:
•f - an integral element of the number field
OUTPUT:
An integral element 𝑔, such that 𝑓 − 𝑔 belongs to the ideal self and such that 𝑔 is a canonical reduced
representative of the coset 𝑓 + 𝐼 (𝐼 =self) as described in the residues function, namely an integral
element with coordinates (𝑟0 , . . . , 𝑟𝑛−1 ), where:
•𝑟𝑖 is reduced modulo 𝑑𝑖
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•𝑑𝑖 = 𝑏𝑖 [𝑖], with 𝑏0 , 𝑏1 , . . . , 𝑏𝑛 HNF basis of the ideal self.
Note: The reduced element 𝑔 is not necessarily small. To get a small 𝑔 use the method small_residue
.
EXAMPLES:
sage:
sage:
sage:
sage:
a^2 sage:
True
k.<a> = NumberField(x^3 + 11)
I = k.ideal(5, a^2 - a + 1)
c = 4*a + 9
I.reduce(c)
2* a
c - I.reduce(c) in I
The reduced element is in the list of canonical representatives returned by the residues method:
sage: I.reduce(c) in list(I.residues())
True
The reduced element does not necessarily have smaller norm (use small_residue for that)
sage: c.norm()
25
sage: (I.reduce(c)).norm()
209
sage: (I.small_residue(c)).norm()
10
Sometimes the canonical reduced representative of 1 won’t be 1 (it depends on the choice of basis for the
ring of integers):
sage: k.<a> = NumberField(x^2 + 23)
sage: I = k.ideal(3)
sage: I.reduce(3*a + 1)
-3/2*a - 1/2
sage: k.ring_of_integers().basis()
[1/2*a + 1/2, a]
AUTHOR: Maite Aranes.
residue_class_degree ( )
Return the residue class degree of this fractional ideal, assuming it is prime. Otherwise, raise a ValueError.
The residue class degree of a prime ideal 𝐼 is the degree of the extension 𝑂𝐾 /𝐼 of its prime subfield.
EXAMPLES:
sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: f = K.factor(19); f
(Fractional ideal (a^2 + a - 3)) * (Fractional ideal (-2*a^4 - a^2 + 2*a ˓→1)) * (Fractional ideal (a^2 + a - 1))
sage: [i.residue_class_degree() for i, _ in f]
[2, 2, 1]
residue_field ( names=None)
Return the residue class field of this fractional ideal, which must be prime.
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EXAMPLES:
sage: K.<a> = NumberField(x^3-7)
sage: P = K.ideal(29).factor()[0][0]
sage: P.residue_field()
Residue field in abar of Fractional ideal (2*a^2 + 3*a - 10)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (2*a^2 + 3*a - 10)
Another example:
sage: K.<a> = NumberField(x^3-7)
sage: P = K.ideal(389).factor()[0][0]; P
Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_class_degree()
2
sage: P.residue_field()
Residue field in abar of Fractional ideal (389, a^2 - 44*a - 9)
sage: P.residue_field('z')
Residue field in z of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF.<w> = P.residue_field()
sage: FF
Residue field in w of Fractional ideal (389, a^2 - 44*a - 9)
sage: FF((a+1)^390)
36
sage: FF(a)
w
An example of reduction maps to the residue field: these are defined on the whole valuation ring, i.e. the
subring of the number field consisting of elements with non-negative valuation. This shows that the issue
raised in trac ticket #1951 has been fixed:
sage: K.<i> = NumberField(x^2 + 1)
sage: P1, P2 = [g[0] for g in K.factor(5)]; (P1,P2)
(Fractional ideal (-i - 2), Fractional ideal (2*i + 1))
sage: a = 1/(1+2*i)
sage: F1, F2 = [g.residue_field() for g in [P1,P2]]; (F1,F2)
(Residue field of Fractional ideal (-i - 2), Residue field of Fractional
˓→ideal (2*i + 1))
sage: a.valuation(P1)
0
sage: F1(i/7)
4
sage: F1(a)
3
sage: a.valuation(P2)
-1
sage: F2(a)
Traceback (most recent call last):
ZeroDivisionError: Cannot reduce field element -2/5*i + 1/5 modulo Fractional
˓→ideal (2*i + 1): it has negative valuation
An example with a relative number field:
sage: L.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: p = L.ideal((-1/2*b - 1/2)*a + 1/2*b - 1/2)
sage: R = p.residue_field(); R
Residue field in abar of Fractional ideal ((-1/2*b - 1/2)*a + 1/2*b - 1/2)
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sage: R.cardinality()
9
sage: R(17)
2
sage: R((a + b)/17)
abar
sage: R(1/b)
2*abar
We verify that trac ticket #8721 is fixed:
sage: L.<a, b> = NumberField([x^2 - 3, x^2 - 5])
sage: L.ideal(a).residue_field()
Residue field in abar of Fractional ideal (a)
residues ( )
Returns a iterator through a complete list of residues modulo this integral ideal.
An error is raised if this fractional ideal is not integral.
OUTPUT:
An iterator through a complete list of residues modulo the integral ideal self. This list is the set of canonical
reduced representatives given by all integral elements with coordinates (𝑟0 , . . . , 𝑟𝑛−1 ), where:
•𝑟𝑖 is reduced modulo 𝑑𝑖
•𝑑𝑖 = 𝑏𝑖 [𝑖], with 𝑏0 , 𝑏1 , . . . , 𝑏𝑛 HNF basis of the ideal.
AUTHOR: John Cremona (modified by Maite Aranes)
EXAMPLES:
sage: K.<i>=NumberField(x^2+1)
sage: res = K.ideal(2).residues(); res
xmrange_iter([[0, 1], [0, 1]], <function <lambda> at 0x...>)
sage: list(res)
[0, i, 1, i + 1]
sage: list(K.ideal(2+i).residues())
[-2*i, -i, 0, i, 2*i]
sage: list(K.ideal(i).residues())
[0]
sage: I = K.ideal(3+6*i)
sage: reps=I.residues()
sage: len(list(reps)) == I.norm()
True
sage: all([r==s or not (r-s) in I for r in reps for s in reps])
˓→(6s on sage.math, 2011)
True
# long time
sage: K.<a> = NumberField(x^3-10)
sage: I = K.ideal(a-1)
sage: len(list(I.residues())) == I.norm()
True
sage: K.<z> = CyclotomicField(11)
sage: len(list(K.primes_above(3)[0].residues())) == 3**5
˓→sage.math, 2011)
True
228
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small_residue ( f )
Given an element 𝑓 of the ambient number field, returns an element 𝑔 such that 𝑓 − 𝑔 belongs to the ideal
self (which must be integral), and 𝑔 is small.
Note: The reduced representative returned is not uniquely determined.
ALGORITHM: Uses Pari function pari:nfeltreduce.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 5)
sage: I = k.ideal(a)
sage: I.small_residue(14)
4
sage:
sage:
sage:
a^2 +
K.<a> = NumberField(x^5 + 7*x^4 + 18*x^2 + x - 3)
I = K.ideal(5)
I.small_residue(a^2 -13)
5* a - 3
class sage.rings.number_field.number_field_ideal. NumberFieldIdeal ( field,
gens,
coerce=True)
Bases: sage.rings.ideal.Ideal_generic
An ideal of a number field.
S_ideal_class_log ( S)
S-class group version of ideal_class_log() .
EXAMPLES:
sage:
sage:
sage:
sage:
[1]
sage:
[3]
K.<a> = QuadraticField(-14)
S = K.primes_above(2)
I = K.ideal(3, a + 1)
I.S_ideal_class_log(S)
I.S_ideal_class_log([])
absolute_norm ( )
A synonym for norm.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).absolute_norm()
5
absolute_ramification_index ( )
A synonym for ramification_index.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + i).absolute_ramification_index()
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artin_symbol ( )
Return the Artin symbol (𝐾/Q, 𝑃 ), where 𝐾 is the number field of 𝑃 =self. This is the unique element
𝑠 of the decomposition group of 𝑃 such that 𝑠(𝑥) = 𝑥𝑝 (mod 𝑃 ) where 𝑝 is the residue characteristic of
𝑃 . (Here 𝑃 (self) should be prime and unramified.)
See the artin_symbol method of the GaloisGroup_v2 class for further documentation and examples.
EXAMPLES:
sage: QuadraticField(-23, 'w').primes_above(7)[0].artin_symbol()
(1,2)
basis ( )
Return a basis for this ideal viewed as a Z -module.
OUTPUT:
An immutable sequence of elements of this ideal (note: their parent is the number field) forming a basis
for this ideal.
EXAMPLES:
sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]
sage: I.basis()
# warning -- choice of basis can be somewhat random
[11, 11*z, 11*z^2, z^3 + 5*z^2 + 4*z + 10, z^4 + z^2 + z + 5, z^5 + z^4 + z^3
˓→+ 2*z^2 + 6*z + 5]
An example of a non-integral ideal.:
sage: J = 1/I
sage: J
# warning -- choice of generators can be somewhat random
Fractional ideal (2/11*z^5 + 2/11*z^4 + 3/11*z^3 + 2/11)
sage: J.basis()
# warning -- choice of basis can be somewhat random
[1, z, z^2, 1/11*z^3 + 7/11*z^2 + 6/11*z + 10/11, 1/11*z^4 + 1/11*z^2 + 1/
˓→11*z + 7/11, 1/11*z^5 + 1/11*z^4 + 1/11*z^3 + 2/11*z^2 + 8/11*z + 7/11]
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: K.ideal(a).basis()
[1, a]
coordinates ( x)
Returns the coordinate vector of 𝑥 with respect to this ideal.
INPUT: x – an element of the number field (or ring of integers) of this ideal.
OUTPUT: List giving the coordinates of 𝑥 with respect to the integral basis of the ideal. In general this
will be a vector of rationals; it will consist of integers if and only if 𝑥 is in the ideal.
AUTHOR: John Cremona 2008-10-31
ALGORITHM:
Uses linear algebra. Provides simpler implementations for _contains_() , is_integral() and
smallest_integer() .
EXAMPLES:
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sage: K.<i> = QuadraticField(-1)
sage: I = K.ideal(7+3*i)
sage: Ibasis = I.integral_basis(); Ibasis
[58, i + 41]
sage: a = 23-14*i
sage: acoords = I.coordinates(a); acoords
(597/58, -14)
sage: sum([Ibasis[j]*acoords[j] for j in range(2)]) == a
True
sage: b = 123+456*i
sage: bcoords = I.coordinates(b); bcoords
(-18573/58, 456)
sage: sum([Ibasis[j]*bcoords[j] for j in range(2)]) == b
True
sage: J = K.ideal(0)
sage: J.coordinates(0)
()
sage: J.coordinates(1)
Traceback (most recent call last):
...
TypeError: vector is not in free module
decomposition_group ( )
Return the decomposition group of self, as a subset of the automorphism group of the number field of self.
Raises an error if the field isn’t Galois. See the decomposition_group method of the GaloisGroup_v2
class for further examples and doctests.
EXAMPLES:
sage: QuadraticField(-23, 'w').primes_above(7)[0].decomposition_group()
Galois group of Number Field in w with defining polynomial x^2 + 23
free_module ( )
Return the free Z-module contained in the vector space associated to the ambient number field, that corresponds to this ideal.
EXAMPLES:
sage: K.<z> = CyclotomicField(7)
sage: I = K.factor(11)[0][0]; I
Fractional ideal (-2*z^4 - 2*z^2 - 2*z + 1)
sage: A = I.free_module()
sage: A
# warning -- choice of basis can be somewhat random
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
[11 0 0 0 0 0]
[ 0 11 0 0 0 0]
[ 0 0 11 0 0 0]
[10 4 5 1 0 0]
[ 5 1 1 0 1 0]
[ 5 6 2 1 1 1]
However, the actual Z-module is not at all random:
sage: A.basis_matrix().change_ring(ZZ).echelon_form()
[ 1 0 0 5 1 1]
[ 0 1 0 1 1 7]
[ 0 0 1 7 6 10]
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[ 0
[ 0
[ 0
0
0
0
0 11 0 0]
0 0 11 0]
0 0 0 11]
The ideal doesn’t have to be integral:
sage: J = I^(-1)
sage: B = J.free_module()
sage: B.echelonized_basis_matrix()
[ 1/11
0
0 7/11 1/11 1/11]
[
0 1/11
0 1/11 1/11 5/11]
[
0
0 1/11 5/11 4/11 10/11]
[
0
0
0
1
0
0]
[
0
0
0
0
1
0]
[
0
0
0
0
0
1]
This also works for relative extensions:
sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: I = K.fractional_ideal(4)
sage: I.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[ 4
0
0
0]
[ -3
7 -1
1]
[ 3
7
1
1]
[ 0 -10
0 -2]
sage: J = I^(-1); J.free_module()
Free module of degree 4 and rank 4 over Integer Ring
User basis matrix:
[ 1/4
0
0
0]
[-3/16 7/16 -1/16 1/16]
[ 3/16 7/16 1/16 1/16]
[
0 -5/8
0 -1/8]
An example of intersecting ideals by intersecting free modules.:
sage: K.<a> = NumberField(x^3 + x^2 - 2*x + 8)
sage: I = K.factor(2)
sage: p1 = I[0][0]; p2 = I[1][0]
sage: N = p1.free_module().intersection(p2.free_module()); N
Free module of degree 3 and rank 3 over Integer Ring
Echelon basis matrix:
[ 1 1/2 1/2]
[ 0
1
1]
[ 0
0
2]
sage: N.index_in(p1.free_module()).abs()
2
gens_reduced ( proof=None)
Express this ideal in terms of at most two generators, and one if possible.
This function indirectly uses bnfisprincipal , so set proof=True if you want to prove correctness
(which is the default).
EXAMPLES:
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sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 5)
sage: K.ideal(0).gens_reduced()
(0,)
sage: J = K.ideal([a+2, 9])
sage: J.gens()
(a + 2, 9)
sage: J.gens_reduced() # random sign
(a + 2,)
sage: K.ideal([a+2, 3]).gens_reduced()
(3, a + 2)
gens_two ( )
Express this ideal using exactly two generators, the first of which is a generator for the intersection of the
ideal with 𝑄.
ALGORITHM: uses PARI’s pari:idealtwoelt function, which runs in randomized polynomial time and is
very fast in practice.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 5)
sage: J = K.ideal([a+2, 9])
sage: J.gens()
(a + 2, 9)
sage: J.gens_two()
(9, a + 2)
sage: K.ideal([a+5, a+8]).gens_two()
(3, a + 2)
sage: K.ideal(0).gens_two()
(0, 0)
The second generator is zero if and only if the ideal is generated by a rational, in contrast to the PARI
function pari:idealtwoelt:
sage: I = K.ideal(12)
sage: pari(K).idealtwoelt(I)
[12, [0, 12]~]
sage: I.gens_two()
(12, 0)
# Note that second element is not zero
ideal_class_log ( proof=None)
Return the output of PARI’s pari:bnfisprincipal for this ideal, i.e. a vector expressing the class of this ideal
in terms of a set of generators for the class group.
Since it uses the PARI method pari:bnfisprincipal, specify proof=True (this is the default setting) to
prove the correctness of the output.
EXAMPLES:
When the class number is 1, the result is always the empty list:
sage: K.<a> = QuadraticField(-163)
sage: J = K.primes_above(random_prime(10^6))[0]
sage: J.ideal_class_log()
[]
An example with class group of order 2. The first ideal is not principal, the second one is:
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sage:
sage:
sage:
[1]
sage:
[0]
K.<a> = QuadraticField(-5)
J = K.ideal(23).factor()[0][0]
J.ideal_class_log()
(J^10).ideal_class_log()
An example with a more complicated class group:
sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 26])
sage: K.class_group()
Class group of order 18 with structure C6 x C3 of Number Field in a with
˓→defining polynomial x^3 - x + 1 over its base field
sage: K.primes_above(7)[0].ideal_class_log() # random
[1, 2]
inertia_group ( )
Return the inertia group of self, i.e. the set of elements s of the Galois group of the number field of self
(which we assume is Galois) such that s acts trivially modulo self. This is the same as the 0th ramification
group of self. See the inertia_group method of the GaloisGroup_v2 class for further examples and
doctests.
EXAMPLES:
sage: QuadraticField(-23, 'w').primes_above(23)[0].inertia_group()
Galois group of Number Field in w with defining polynomial x^2 + 23
integral_basis ( )
Return a list of generators for this ideal as a Z-module.
EXAMPLES:
sage:
sage:
sage:
sage:
[2, i
R.<x> = PolynomialRing(QQ)
K.<i> = NumberField(x^2 + 1)
J = K.ideal(i+1)
J.integral_basis()
+ 1]
integral_split ( )
Return a tuple (𝐼, 𝑑), where 𝐼 is an integral ideal, and 𝑑 is the smallest positive integer such that this ideal
is equal to 𝐼/𝑑.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2-5)
sage: I = K.ideal(2/(5+a))
sage: I.is_integral()
False
sage: J,d = I.integral_split()
sage: J
Fractional ideal (-1/2*a + 5/2)
sage: J.is_integral()
True
sage: d
5
sage: I == J/d
True
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intersection ( other)
Return the intersection of self and other.
EXAMPLES:
sage: K.<a> = QuadraticField(-11)
sage: p = K.ideal((a + 1)/2); q = K.ideal((a + 3)/2)
sage: p.intersection(q) == q.intersection(p) == K.ideal(a-2)
True
An example with non-principal ideals:
sage:
sage:
sage:
sage:
True
L.<a> = NumberField(x^3 - 7)
p = L.ideal(a^2 + a + 1, 2)
q = L.ideal(a+1)
p.intersection(q) == L.ideal(8, 2*a + 2)
A relative example:
sage:
sage:
sage:
sage:
True
L.<a,b> = NumberField([x^2 + 11, x^2 - 5])
A = L.ideal([15, (-3/2*b + 7/2)*a - 8])
B = L.ideal([6, (-1/2*b + 1)*a - b - 5/2])
A.intersection(B) == L.ideal(-1/2*a - 3/2*b - 1)
is_integral ( )
Return True if this ideal is integral.
EXAMPLES:
sage:
sage:
sage:
True
sage:
False
R.<x> = PolynomialRing(QQ)
K.<a> = NumberField(x^5-x+1)
K.ideal(a).is_integral()
(K.ideal(1) / (3*a+1)).is_integral()
is_maximal ( )
Return True if this ideal is maximal. This is equivalent to self being prime and nonzero.
EXAMPLES:
sage: K.<a> = NumberField(x^3 + 3); K
Number Field in a with defining polynomial x^3 + 3
sage: K.ideal(5).is_maximal()
False
sage: K.ideal(7).is_maximal()
True
is_prime ( )
Return True if this ideal is prime.
EXAMPLES:
sage: K.<a> = NumberField(x^2 - 17); K
Number Field in a with defining polynomial x^2 - 17
sage: K.ideal(5).is_prime()
# inert prime
True
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sage: K.ideal(13).is_prime()
False
sage: K.ideal(17).is_prime()
False
# split
# ramified
is_principal ( proof=None)
Return True if this ideal is principal.
Since it uses the PARI method pari:bnfisprincipal, specify proof=True (this is the default setting) to
prove the correctness of the output.
EXAMPLES:
sage: K = QuadraticField(-119,’a’) sage: P = K.factor(2)[1][0] sage: P.is_principal() False sage:
I = P^5 sage: I.is_principal() True sage: I # random Fractional ideal (-1/2*a + 3/2) sage: P =
K.ideal([2]).factor()[1][0] sage: I = P^5 sage: I.is_principal() True
is_zero ( )
Return True iff self is the zero ideal
Note that (0) is a NumberFieldIdeal , not a NumberFieldFractionalIdeal .
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 2); K
Number Field in a with defining polynomial x^2 + 2
sage: K.ideal(3).is_zero()
False
sage: I=K.ideal(0); I.is_zero()
True
sage: I
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
norm ( )
Return the norm of this fractional ideal as a rational number.
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 23); K
Number Field in a with defining polynomial x^4 + 23
sage: I = K.ideal(19); I
Fractional ideal (19)
sage: factor(I.norm())
19^4
sage: F = I.factor()
sage: F[0][0].norm().factor()
19^2
number_field ( )
Return the number field that this is a fractional ideal in.
EXAMPLES:
sage: K.<a> = NumberField(x^2 +
Number Field in a with defining
sage: K.ideal(3).number_field()
Number Field in a with defining
sage: K.ideal(0).number_field()
Number Field in a with defining
236
2); K
polynomial x^2 + 2
polynomial x^2 + 2
# not tested (not implemented)
polynomial x^2 + 2
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pari_hnf ( )
Return PARI’s representation of this ideal in Hermite normal form.
EXAMPLES:
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^3 - 2)
sage: I = K.ideal(2/(5+a))
sage: I.pari_hnf()
[2, 0, 50/127; 0, 2, 244/127; 0, 0, 2/127]
pari_prime ( )
Returns a PARI prime ideal corresponding to the ideal self .
INPUT:
•self - a prime ideal.
OUTPUT: a PARI “prime ideal”, i.e. a five-component vector [𝑝, 𝑎, 𝑒, 𝑓, 𝑏] representing the prime ideal
𝑝𝑂𝐾 + 𝑎𝑂𝐾 , 𝑒, 𝑓 as usual, 𝑎 as vector of components on the integral basis, 𝑏 Lenstra’s constant.
EXAMPLES:
sage: K.<i> = QuadraticField(-1)
sage: K.ideal(3).pari_prime()
[3, [3, 0]~, 1, 2, 1]
sage: K.ideal(2+i).pari_prime()
[5, [2, 1]~, 1, 1, [-2, -1; 1, -2]]
sage: K.ideal(2).pari_prime()
Traceback (most recent call last):
...
ValueError: Fractional ideal (2) is not a prime ideal
ramification_group ( v)
Return the 𝑣‘th ramification group of self, i.e. the set of elements 𝑠 of the Galois group of the number field
of self (which we assume is Galois) such that 𝑠 acts trivially modulo the (𝑣 + 1)‘st power of self. See the
ramification_group method of the GaloisGroup class for further examples and doctests.
EXAMPLES:
sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(0)
Galois group of Number Field in w with defining polynomial x^2 + 23
sage: QuadraticField(-23, 'w').primes_above(23)[0].ramification_group(1)
Subgroup [()] of Galois group of Number Field in w with defining polynomial x^
˓→2 + 23
random_element ( *args, **kwds)
Return a random element of this order.
INPUT:
•*args , *kwds - Parameters passed to the random integer function. See the documentation of
ZZ.random_element() for details.
OUTPUT:
A random element of this fractional ideal, computed as a random Z-linear combination of the basis.
EXAMPLES:
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sage: K.<a> = NumberField(x^3 + 2)
sage: I = K.ideal(1-a)
sage: I.random_element() # random output
-a^2 - a - 19
sage: I.random_element(distribution="uniform") # random output
a^2 - 2*a - 8
sage: I.random_element(-30,30) # random output
-7*a^2 - 17*a - 75
sage: I.random_element(-100, 200).is_integral()
True
sage: I.random_element(-30,30).parent() is K
True
A relative example:
sage: K.<a, b> = NumberField([x^2 + 2, x^2 + 1000*x + 1])
sage: I = K.ideal(1-a)
sage: I.random_element() # random output
17/500002*a^3 + 737253/250001*a^2 - 1494505893/500002*a + 752473260/250001
sage: I.random_element().is_integral()
True
sage: I.random_element(-100, 200).parent() is K
True
reduce_equiv ( )
Return a small ideal that is equivalent to self in the group of fractional ideals modulo principal ideals. Very
often (but not always) if self is principal then this function returns the unit ideal.
ALGORITHM: Calls pari’s idealred function.
EXAMPLES:
sage: K.<w> = NumberField(x^2 + 23)
sage: I = ideal(w*23^5); I
Fractional ideal (6436343*w)
sage: I.reduce_equiv()
Fractional ideal (1)
sage: I = K.class_group().0.ideal()^10; I
Fractional ideal (1024, 1/2*w + 979/2)
sage: I.reduce_equiv()
Fractional ideal (2, 1/2*w - 1/2)
relative_norm ( )
A synonym for norm.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + 2*i).relative_norm()
5
relative_ramification_index ( )
A synonym for ramification_index.
EXAMPLES:
sage: K.<i> = NumberField(x^2 + 1)
sage: K.ideal(1 + i).relative_ramification_index()
2
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residue_symbol ( e, m, check=True)
The m-th power residue symbol for an element e and the proper ideal.
(︁ 𝛼 )︁
𝑁 (P)−1
≡ 𝛼 𝑚 mod P
P
Note: accepts m=1, in which case returns 1
Note: can also be called for an element from sage.rings.number_field_element.residue_symbol
Note: e is coerced into the number field of self
Note: if m=2, e is an integer, and self.number_field() has absolute degree 1 (i.e. it is a copy of the
rationals), then this calls kronecker_symbol, which is implemented using GMP.
INPUT:
•e - element of the number field
•m - positive integer
OUTPUT:
•an m-th root of unity in the number field
EXAMPLES:
Quadratic Residue (7 is not a square modulo 11):
sage: K.<a> = NumberField(x - 1)
sage: K.ideal(11).residue_symbol(7,2)
-1
Cubic Residue:
sage: K.<w> = NumberField(x^2 - x + 1)
sage: K.ideal(17).residue_symbol(w^2 + 3,3)
-w
The field must contain the m-th roots of unity:
sage: K.<w> = NumberField(x^2 - x + 1)
sage: K.ideal(17).residue_symbol(w^2 + 3,5)
Traceback (most recent call last):
...
ValueError: The residue symbol to that power is not defined for the number
˓→field
smallest_integer ( )
Return the smallest non-negative integer in 𝐼 ∩ Z, where 𝐼 is this ideal. If 𝐼 = 0, returns 0.
EXAMPLES:
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sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2+6)
sage: I = K.ideal([4,a])/7; I
Fractional ideal (2/7, 1/7*a)
sage: I.smallest_integer()
2
valuation ( p)
Return the valuation of self at p .
INPUT:
•p – a prime ideal p of this number field.
OUTPUT:
(integer) The valuation of this fractional ideal at the prime p. If p is not prime, raise a ValueError.
EXAMPLES:
sage: K.<a> = NumberField(x^5 + 2); K
Number Field in a with defining polynomial x^5 + 2
sage: i = K.ideal(38); i
Fractional ideal (38)
sage: i.valuation(K.factor(19)[0][0])
1
sage: i.valuation(K.factor(2)[0][0])
5
sage: i.valuation(K.factor(3)[0][0])
0
sage: i.valuation(0)
Traceback (most recent call last):
...
ValueError: p (= Ideal (0) of Number Field in a with defining polynomial x^5
˓→+ 2) must be nonzero
sage: K.ideal(0).valuation(K.factor(2)[0][0])
+Infinity
class sage.rings.number_field.number_field_ideal. QuotientMap ( K, M_OK_change,
Q, I)
Class to hold data needed by quotient maps from number field orders to residue fields. These are only partial
maps: the exact domain is the appropriate valuation ring. For examples, see residue_field() .
sage.rings.number_field.number_field_ideal. basis_to_module ( B, K)
Given a basis 𝐵 of elements for a Z-submodule of a number field 𝐾, return the corresponding Z-submodule.
EXAMPLES:
sage: K.<w> = NumberField(x^4 + 1)
sage: from sage.rings.number_field.number_field_ideal import basis_to_module
sage: basis_to_module([K.0, K.0^2 + 3], K)
Free module of degree 4 and rank 2 over Integer Ring
User basis matrix:
[0 1 0 0]
[3 0 1 0]
sage.rings.number_field.number_field_ideal. is_NumberFieldFractionalIdeal ( x)
Return True if x is a fractional ideal of a number field.
EXAMPLES:
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sage: from sage.rings.number_field.number_field_ideal import is_
˓→NumberFieldFractionalIdeal
sage: is_NumberFieldFractionalIdeal(2/3)
False
sage: is_NumberFieldFractionalIdeal(ideal(5))
False
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldFractionalIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldFractionalIdeal(Z)
False
sage.rings.number_field.number_field_ideal. is_NumberFieldIdeal ( x)
Return True if x is an ideal of a number field.
EXAMPLES:
sage: from sage.rings.number_field.number_field_ideal import is_NumberFieldIdeal
sage: is_NumberFieldIdeal(2/3)
False
sage: is_NumberFieldIdeal(ideal(5))
False
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldIdeal(I)
True
sage: Z = k.ideal(0); Z
Ideal (0) of Number Field in a with defining polynomial x^2 + 2
sage: is_NumberFieldIdeal(Z)
True
sage.rings.number_field.number_field_ideal. quotient_char_p ( I, p)
Given an integral ideal 𝐼 that contains a prime number 𝑝, compute a vector space 𝑉 = (𝑂𝐾 mod 𝑝)/(𝐼
mod 𝑝), along with a homomorphism 𝑂𝐾 → 𝑉 and a section 𝑉 → 𝑂𝐾 .
EXAMPLES:
sage: from sage.rings.number_field.number_field_ideal import quotient_char_p
sage: K.<i> = NumberField(x^2 + 1); O = K.maximal_order(); I = K.fractional_
˓→ideal(15)
sage: quotient_char_p(I, 5)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 5 where
V: Vector space of dimension 2 over Finite Field of size 5
W: Vector space of degree 2 and dimension 0 over Finite Field of size 5
Basis matrix:
[]
sage: quotient_char_p(I, 3)[0]
Vector space quotient V/W of dimension 2 over Finite Field of size 3 where
V: Vector space of dimension 2 over Finite Field of size 3
W: Vector space of degree 2 and dimension 0 over Finite Field of size 3
Basis matrix:
[]
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sage: I = K.factor(13)[0][0]; I
Fractional ideal (-3*i - 2)
sage: I.residue_class_degree()
1
sage: quotient_char_p(I, 13)[0]
Vector space quotient V/W of dimension 1 over Finite Field of size 13 where
V: Vector space of dimension 2 over Finite Field of size 13
W: Vector space of degree 2 and dimension 1 over Finite Field of size 13
Basis matrix:
[1 8]
3.3 Relative Number Field Ideals
AUTHORS:
• Steven Sivek (2005-05-16)
• William Stein (2007-09-06)
• Nick Alexander (2009-01)
EXAMPLES:
sage: K.<a,b> = NumberField([x^2 + 1, x^2 + 2])
sage: A = K.absolute_field('z')
sage: I = A.factor(7)[0][0]
sage: from_A, to_A = A.structure()
sage: G = [from_A(z) for z in I.gens()]; G
[7, -2*b*a - 1]
sage: K.fractional_ideal(G)
Fractional ideal (2*b*a + 1)
sage: K.fractional_ideal(G).absolute_norm().factor()
7^2
class sage.rings.number_field.number_field_ideal_rel. NumberFieldFractionalIdeal_rel ( field,
gens,
coerce=True)
Bases: sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal
An ideal of a relative number field.
EXAMPLES:
sage: K.<a> = NumberField([x^2 + 1, x^2 + 2]); K
Number Field in a0 with defining polynomial x^2 + 1 over its base field
sage: i = K.ideal(38); i
Fractional ideal (38)
sage: K.<a0, a1> = NumberField([x^2 + 1, x^2 + 2]); K
Number Field in a0 with defining polynomial x^2 + 1 over its base field
sage: i = K.ideal([a0+1]); i # random
Fractional ideal (-a1*a0)
sage: (g, ) = i.gens_reduced(); g # random
-a1*a0
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sage: (g / (a0 + 1)).is_integral()
True
sage: ((a0 + 1) / g).is_integral()
True
absolute_ideal ( names=’a’)
If this is an ideal in the extension 𝐿/𝐾, return the ideal with the same generators in the absolute field 𝐿/Q.
INPUT:
•names (optional) – string; name of generator of the absolute field
EXAMPLES:
sage:
sage:
sage:
sage:
x = ZZ['x'].0
K.<b> = NumberField(x^2 - 2)
L.<c> = K.extension(x^2 - b)
F.<m> = L.absolute_field()
An example of an inert ideal:
sage: P = F.factor(13)[0][0]; P
Fractional ideal (13)
sage: J = L.ideal(13)
sage: J.absolute_ideal()
Fractional ideal (13)
Now a non-trivial ideal in 𝐿 that is principal in the subfield 𝐾. Since the optional ‘names’ argument is not
passed, the generators of the absolute ideal J are returned in terms of the default field generator ‘a’. This
does not agree with the generator ‘m’ of the absolute field F defined above:
sage: J = L.ideal(b); J
Fractional ideal (b)
sage: J.absolute_ideal()
Fractional ideal (a^2)
sage: J.relative_norm()
Fractional ideal (2)
sage: J.absolute_norm()
4
sage: J.absolute_ideal().norm()
4
Now pass ‘m’ as the name for the generator of the absolute field:
sage: J.absolute_ideal(‘m’) Fractional ideal (m^2)
Now an ideal not generated by an element of 𝐾:
sage: J = L.ideal(c); J
Fractional ideal (c)
sage: J.absolute_ideal()
Fractional ideal (a)
sage: J.absolute_norm()
2
sage: J.ideal_below()
Fractional ideal (b)
sage: J.ideal_below().norm()
2
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absolute_norm ( )
Compute the absolute norm of this fractional ideal in a relative number field, returning a positive integer.
EXAMPLES:
sage: L.<a, b, c> = QQ.extension([x^2 - 23, x^2 - 5, x^2 - 7])
sage: I = L.ideal(a + b)
sage: I.absolute_norm()
104976
sage: I.relative_norm().relative_norm().relative_norm()
104976
absolute_ramification_index ( )
Return the absolute ramification index of this fractional ideal, assuming it is prime. Otherwise, raise a
ValueError.
The absolute ramification index is the power of this prime appearing in the factorization of the rational
prime that this prime lies over.
Use relative_ramification_index to obtain the power of this prime occurring in the factorization of the
prime ideal of the base field that this prime lies over.
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
sage:
4
sage:
3
sage:
True
PQ.<X> = QQ[]
F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
PF.<Y> = F[]
K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
I = K.ideal(3, c)
I.absolute_ramification_index()
I.smallest_integer()
K.ideal(3) == I^4
element_1_mod ( other)
Returns an element 𝑟 in this ideal such that 1 − 𝑟 is in other.
An error is raised if either ideal is not integral of if they are not coprime.
INPUT:
•other – another ideal of the same field, or generators of an ideal.
OUTPUT:
an element 𝑟 of the ideal self such that 1 − 𝑟 is in the ideal other.
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
True
sage:
True
K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
I = Ideal(2, (a - 3*b + 2)/2)
J = K.ideal(a)
z = I.element_1_mod(J)
z in I
1 - z in J
factor ( )
Factor the ideal by factoring the corresponding ideal in the absolute number field.
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EXAMPLES:
sage: K.<a, b> = QQ.extension([x^2 + 11,
sage: K.factor(5)
(Fractional ideal (5, (-1/4*b - 1/4)*a +
˓→(5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
sage: K.ideal(5).factor()
(Fractional ideal (5, (-1/4*b - 1/4)*a +
˓→(5, (-1/4*b - 1/4)*a + 1/4*b - 7/4))^2
sage: K.ideal(5).prime_factors()
[Fractional ideal (5, (-1/4*b - 1/4)*a +
Fractional ideal (5, (-1/4*b - 1/4)*a +
x^2 - 5])
1/4*b - 3/4))^2 * (Fractional ideal
1/4*b - 3/4))^2 * (Fractional ideal
1/4*b - 3/4),
1/4*b - 7/4)]
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(c)
sage: P = K.ideal((b*a - b - 1)*c/2 + a - 1)
sage: Q = K.ideal((b*a - b - 1)*c/2)
sage: list(I.factor()) == [(P, 2), (Q, 1)]
True
sage: I == P^2*Q
True
sage: [p.is_prime() for p in [P, Q]]
[True, True]
free_module ( )
Return this ideal as a Z-submodule of the Q-vector space corresponding to the ambient number field.
EXAMPLES:
sage: K.<a, b> = NumberField([x^3 - x + 1, x^2 + 23])
sage: I = K.ideal(a*b - 1)
sage: I.free_module()
Free module of degree 6 and rank 6 over Integer Ring
User basis matrix:
...
sage: I.free_module().is_submodule(K.maximal_order().free_module())
True
gens_reduced ( )
Return a small set of generators for this ideal. This will always return a single generator if one exists (i.e.
if the ideal is principal), and otherwise two generators.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: I = K.ideal((a + 1)*b/2 + 1)
sage: I.gens_reduced()
(1/2*b*a + 1/2*b + 1,)
ideal_below ( )
Compute the ideal of 𝐾 below this ideal of 𝐿.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2+6)
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sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: N = L.ideal(b)
sage: M = N.ideal_below(); M == K.ideal([-a])
True
sage: Np = L.ideal( [ L(t) for t in M.gens() ])
sage: Np.ideal_below() == M
True
sage: M.parent()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 6
sage: M.ring()
Number Field in a with defining polynomial x^2 + 6
sage: M.ring() is K
True
This example concerns an inert ideal:
sage: K = NumberField(x^4 + 6*x^2 + 24, 'a')
sage: K.factor(7)
Fractional ideal (7)
sage: K0, K0_into_K, _ = K.subfields(2)[0]
sage: K0
Number Field in a0 with defining polynomial x^2 - 6*x + 24
sage: L = K.relativize(K0_into_K, 'c'); L
Number Field in c with defining polynomial x^2 + a0 over its base field
sage: L.base_field() is K0
True
sage: L.ideal(7)
Fractional ideal (7)
sage: L.ideal(7).ideal_below()
Fractional ideal (7)
sage: L.ideal(7).ideal_below().number_field() is K0
True
This example concerns an ideal that splits in the quadratic field but each factor ideal remains inert in the
extension:
sage: len(K.factor(19))
2
sage: K0 = L.base_field(); a0 = K0.gen()
sage: len(K0.factor(19))
2
sage: w1 = -a0 + 1; P1 = K0.ideal([w1])
sage: P1.norm().factor(), P1.is_prime()
(19, True)
sage: L_into_K, K_into_L = L.structure()
sage: L.ideal(K_into_L(K0_into_K(w1))).ideal_below() == P1
True
The choice of embedding of quadratic field into quartic field matters:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
246
rho, tau = K0.embeddings(K)
L1 = K.relativize(rho, 'b')
L2 = K.relativize(tau, 'b')
L1_into_K, K_into_L1 = L1.structure()
L2_into_K, K_into_L2 = L2.structure()
a = K.gen()
P = K.ideal([a^2 + 5])
K_into_L1(P).ideal_below() == K0.ideal([-a0 + 1])
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True
sage: K_into_L2(P).ideal_below() == K0.ideal([-a0 + 5])
True
sage: K0.ideal([-a0 + 1]) == K0.ideal([-a0 + 5])
False
It works when the base_field is itself a relative number field:
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(3, c)
sage: J = I.ideal_below(); J
Fractional ideal (-b)
sage: J.number_field() == F
True
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.ideal_below()
Fractional ideal (-6*a + 2)
integral_basis ( )
Return a basis for self as a Z-module.
EXAMPLES:
sage:
sage:
sage:
[438,
K.<a,b> = NumberField([x^2 + 1, x^2 - 3])
I = K.ideal(17*b - 3*a)
x = I.integral_basis(); x # random
-b*a + 309, 219*a - 219*b, 156*a - 154*b]
The exact results are somewhat unpredictable, hence the # random flag, but we can test that they are
indeed a basis:
sage: V, _, phi = K.absolute_vector_space()
sage: V.span([phi(u) for u in x], ZZ) == I.free_module()
True
integral_split ( )
Return a tuple (𝐼, 𝑑), where 𝐼 is an integral ideal, and 𝑑 is the smallest positive integer such that this ideal
is equal to 𝐼/𝑑.
EXAMPLES:
sage:
sage:
sage:
sage:
True
sage:
True
K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
I = K.ideal([a + b/3])
J, d = I.integral_split()
J.is_integral()
J == d*I
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is_integral ( )
Return True if this ideal is integral.
EXAMPLES:
sage:
sage:
sage:
True
sage:
False
K.<a, b> = QQ.extension([x^2 + 11, x^2 - 5])
I = K.ideal(7).prime_factors()[0]
I.is_integral()
(I/2).is_integral()
is_prime ( )
Return True if this ideal of a relative number field is prime.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
sage: K.ideal(a + b).is_prime()
True
sage: K.ideal(13).is_prime()
False
is_principal ( proof=None)
Return True if this ideal is principal. If so, set self.__reduced_generators, with length one.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 - 23, x^2 + 1])
sage: I = K.ideal([7, (-1/2*b - 3/2)*a + 3/2*b + 9/2])
sage: I.is_principal()
True
sage: I # random
Fractional ideal ((1/2*b + 1/2)*a - 3/2*b - 3/2)
is_zero ( )
Return True if this is the zero ideal.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 3, x^3 + 4])
sage: K.ideal(17).is_zero()
False
sage: K.ideal(0).is_zero()
True
norm ( )
The norm of a fractional ideal in a relative number field is deliberately unimplemented, so that a user
cannot mistake the absolute norm for the relative norm, or vice versa.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: K.ideal(2).norm()
Traceback (most recent call last):
...
NotImplementedError: For a fractional ideal in a relative number field you
˓→must use relative_norm or absolute_norm as appropriate
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pari_rhnf ( )
Return PARI’s representation of this relative ideal in Hermite normal form.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 23, x^2 - 7])
sage: I = K.ideal(2, (a + 2*b + 3)/2)
sage: I.pari_rhnf()
[[1, -2; 0, 1], [[2, 1; 0, 1], 1/2]]
ramification_index ( )
For ideals in relative number fields, ramification_index
is deliberately not implemented in order to avoid ambiguity.
Either relative_ramification_index()
or
absolute_ramification_index() should be used instead.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 + 1, x^2 - 2])
sage: K.ideal(2).ramification_index()
Traceback (most recent call last):
...
NotImplementedError: For an ideal in a relative number field you must use
˓→relative_ramification_index or absolute_ramification_index as appropriate
relative_norm ( )
Compute the relative norm of this fractional ideal in a relative number field, returning an ideal in the base
field.
EXAMPLES:
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2+6)
sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: N = L.ideal(b).relative_norm(); N
Fractional ideal (-a)
sage: N.parent()
Monoid of ideals of Number Field in a with defining polynomial x^2 + 6
sage: N.ring()
Number Field in a with defining polynomial x^2 + 6
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: K.ideal(1).relative_norm()
Fractional ideal (1)
sage: K.ideal(13).relative_norm().relative_norm()
Fractional ideal (28561)
sage: K.ideal(13).relative_norm().relative_norm().relative_norm()
815730721
sage: K.ideal(13).absolute_norm()
815730721
Number fields defined by non-monic and non-integral polynomials are supported (trac ticket #252):
sage: K.<a> = NumberField(2*x^2 - 1/3)
sage: L.<b> = K.extension(5*x^2 + 1)
sage: P = L.primes_above(2)[0]
sage: P.relative_norm()
Fractional ideal (-6*a + 2)
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relative_ramification_index ( )
Return the relative ramification index of this fractional ideal, assuming it is prime. Otherwise, raise a
ValueError.
The relative ramification index is the power of this prime appearing in the factorization of the prime ideal
of the base field that this prime lies over.
Use absolute_ramification_index to obtain the power of this prime occurring in the factorization of the
rational prime that this prime lies over.
EXAMPLES:
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: I = K.ideal(3, c)
sage: I.relative_ramification_index()
2
sage: I.ideal_below() # random sign
Fractional ideal (b)
sage: I.ideal_below() == K.ideal(b)
True
sage: K.ideal(b) == I^2
True
residue_class_degree ( )
Return the residue class degree of this prime.
EXAMPLES:
sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberFieldTower([X^2 - 2, X^2 - 3])
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a + b)*a*b)
sage: [I.residue_class_degree() for I in K.ideal(c).prime_factors()]
[1, 2]
residues ( )
Returns a iterator through a complete list of residues modulo this integral ideal.
An error is raised if this fractional ideal is not integral.
EXAMPLES:
sage: K.<a, w> = NumberFieldTower([x^2 - 3, x^2 + x + 1])
sage: I = K.ideal(6, -w*a - w + 4)
sage: list(I.residues())[:5]
[(25/3*w - 1/3)*a + 22*w + 1,
(16/3*w - 1/3)*a + 13*w,
(7/3*w - 1/3)*a + 4*w - 1,
(-2/3*w - 1/3)*a - 5*w - 2,
(-11/3*w - 1/3)*a - 14*w - 3]
smallest_integer ( )
Return the smallest non-negative integer in 𝐼 ∩ Z, where 𝐼 is this ideal. If 𝐼 = 0, returns 0.
EXAMPLES:
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sage: K.<a, b> = NumberFieldTower([x^2 - 23, x^2 + 1])
sage: I = K.ideal([a + b])
sage: I.smallest_integer()
12
sage: [m for m in range(13) if m in I]
[0, 12]
valuation ( p)
Return the valuation of this fractional ideal at p .
INPUT:
•p – a prime ideal p of this relative number field.
OUTPUT:
(integer) The valuation of this fractional ideal at the prime p. If p is not prime, raise a ValueError.
EXAMPLES:
sage: K.<a, b> = NumberField([x^2 - 17, x^3 - 2])
sage: A = K.ideal(a + b)
sage: A.is_prime()
True
sage: (A*K.ideal(3)).valuation(A)
1
sage: K.ideal(25).valuation(5)
Traceback (most recent call last):
...
ValueError: p (= Fractional ideal (5)) must be a prime
sage.rings.number_field.number_field_ideal_rel. is_NumberFieldFractionalIdeal_rel ( x)
Return True if x is a fractional ideal of a relative number field.
EXAMPLES:
sage: from sage.rings.number_field.number_field_ideal_rel import is_
˓→NumberFieldFractionalIdeal_rel
sage: from sage.rings.number_field.number_field_ideal import is_
˓→NumberFieldFractionalIdeal
sage: is_NumberFieldFractionalIdeal_rel(2/3)
False
sage: is_NumberFieldFractionalIdeal_rel(ideal(5))
False
sage: k.<a> = NumberField(x^2 + 2)
sage: I = k.ideal([a + 1]); I
Fractional ideal (a + 1)
sage: is_NumberFieldFractionalIdeal_rel(I)
False
sage: R.<x> = QQ[]
sage: K.<a> = NumberField(x^2+6)
sage: L.<b> = K.extension(K['x'].gen()^4 + a)
sage: I = L.ideal(b); I
Fractional ideal (6, b)
sage: is_NumberFieldFractionalIdeal_rel(I)
True
sage: N = I.relative_norm(); N
Fractional ideal (-a)
sage: is_NumberFieldFractionalIdeal_rel(N)
False
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sage: is_NumberFieldFractionalIdeal(N)
True
3.4 Class Groups of Number Fields
An element of a class group is stored as a pair consisting of both an explicit ideal in that ideal class, and a list of
exponents giving that ideal class in terms of the generators of the parent class group. These can be accessed with the
ideal() and exponents() methods respectively.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 23)
sage: I = K.class_group().gen(); I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I.ideal()
Fractional ideal (2, 1/2*a - 1/2)
sage: I.exponents()
(1,)
sage: I.ideal() * I.ideal()
Fractional ideal (4, 1/2*a + 3/2)
sage: (I.ideal() * I.ideal()).reduce_equiv()
Fractional ideal (2, 1/2*a + 1/2)
sage: J = I * I; J
# class group multiplication is automatically reduced
Fractional ideal class (2, 1/2*a + 1/2)
sage: J.ideal()
Fractional ideal (2, 1/2*a + 1/2)
sage: J.exponents()
(2,)
sage: I * I.ideal()
# ideal classes coerce to their representative ideal
Fractional ideal (4, 1/2*a + 3/2)
sage: O = K.OK(); O
Maximal Order in Number Field in a with defining polynomial x^2 + 23
sage: O*(2, 1/2*a + 1/2)
Fractional ideal (2, 1/2*a + 1/2)
sage: (O*(2, 1/2*a + 1/2)).is_principal()
False
sage: (O*(2, 1/2*a + 1/2))^3
Fractional ideal (1/2*a - 3/2)
class sage.rings.number_field.class_group. ClassGroup ( gens_orders,
names,
number_field, gens, proof=True)
Bases: sage.groups.abelian_gps.values.AbelianGroupWithValues_class
The class group of a number field.
EXAMPLES:
sage: K.<a> = NumberField(x^2 + 23)
sage: G = K.class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining
˓→polynomial x^2 + 23
sage: G.category()
Category of finite enumerated commutative groups
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Note the distinction between abstract generators, their ideal, and exponents:
sage: C = NumberField(x^2 + 120071, 'a').class_group(); C
Class group of order 500 with structure C250 x C2
of Number Field in a with defining polynomial x^2 + 120071
sage: c = C.gen(0)
sage: c # random
Fractional ideal class (5, 1/2*a + 3/2)
sage: c.ideal() # random
Fractional ideal (5, 1/2*a + 3/2)
sage: c.ideal() is c.value()
# alias
True
sage: c.exponents()
(1, 0)
Element
alias of FractionalIdealClass
gens_ideals ( )
Return generating ideals for the (S-)class group.
This is an alias for gens_values() .
OUTPUT:
A tuple of ideals, one for each abstract Abelian group generator.
EXAMPLES:
sage: K.<a> = NumberField(x^4 + 23)
sage: K.class_group().gens_ideals()
# random gens (platform dependent)
(Fractional ideal (2, 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/4),)
sage: C = NumberField(x^2 + x + 23899, 'a').class_group(); C
Class group of order 68 with structure C34 x C2 of Number Field
in a with defining polynomial x^2 + x + 23899
sage: C.gens()
(Fractional ideal class (7, a + 5), Fractional ideal class (5, a + 3))
sage: C.gens_ideals()
(Fractional ideal (7, a + 5), Fractional ideal (5, a + 3))
number_field ( )
Return the number field that this (S-)class group is attached to.
EXAMPLES:
sage: C = NumberField(x^2 + 23, 'w').class_group(); C
Class group of order 3 with structure C3 of Number Field in w with defining
˓→polynomial x^2 + 23
sage: C.number_field()
Number Field in w with defining polynomial x^2 + 23
sage: K.<a> = QuadraticField(-14)
sage: CS = K.S_class_group(K.primes_above(2))
sage: CS.number_field()
Number Field in a with defining polynomial x^2 + 14
class sage.rings.number_field.class_group. FractionalIdealClass ( parent, element,
ideal=None)
Bases: sage.groups.abelian_gps.values.AbelianGroupWithValuesElement
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A fractional ideal class in a number field.
EXAMPLES:
sage: G = NumberField(x^2 + 23,'a').class_group(); G
Class group of order 3 with structure C3 of Number Field in a with defining
˓→polynomial x^2 + 23
sage: I = G.0; I
Fractional ideal class (2, 1/2*a - 1/2)
sage: I.ideal()
Fractional ideal (2, 1/2*a - 1/2)
EXAMPLES::
sage: K.<w>=QuadraticField(-23)
sage: OK=K.ring_of_integers()
sage: C=OK.class_group()
sage: P2a,P2b=[P for P,e in (2*OK).factor()]
sage: c = C(P2a); c
Fractional ideal class (2, 1/2*w - 1/2)
sage: c.gens()
(2, 1/2*w - 1/2)
gens ( )
Return generators for a representative ideal in this (S-)ideal class.
EXAMPLES:
sage: K.<w>=QuadraticField(-23)
sage: OK = K.ring_of_integers()
sage: C = OK.class_group()
sage: P2a,P2b=[P for P,e in (2*OK).factor()]
sage: c = C(P2a); c
Fractional ideal class (2, 1/2*w - 1/2)
sage: c.gens()
(2, 1/2*w - 1/2)
ideal ( )
Return a representative ideal in this ideal class.
EXAMPLES:
sage: K.<w>=QuadraticField(-23)
sage: OK=K.ring_of_integers()
sage: C=OK.class_group()
sage: P2a,P2b=[P for P,e in (2*OK).factor()]
sage: c=C(P2a); c
Fractional ideal class (2, 1/2*w - 1/2)
sage: c.ideal()
Fractional ideal (2, 1/2*w - 1/2)
inverse ( )
Return the multiplicative inverse of this ideal class.
EXAMPLES:
sage: K.<a> = NumberField(x^3 - 3*x + 8); G = K.class_group()
sage: G(2, a).inverse()
Fractional ideal class (2, a^2 + 2*a - 1)
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sage: ~G(2, a)
Fractional ideal class (2, a^2 + 2*a - 1)
is_principal ( )
Returns True iff this ideal class is the trivial (principal) class
EXAMPLES:
sage:
sage:
sage:
sage:
sage:
sage:
False
sage:
False
sage:
True
K.<w>=QuadraticField(-23)
OK=K.ring_of_integers()
C=OK.class_group()
P2a,P2b=[P for P,e in (2*OK).factor()]
c=C(P2a)
c.is_principal()
(c^2).is_principal()
(c^3).is_principal()
reduce ( )
Return representative for this ideal class that has been reduced using PARI’s idealred.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 20072); G = k.class_group(); G
Class group of order 76 with structure C38 x C2
of Number Field in a with defining polynomial x^2 + 20072
sage: I = (G.0)^11; I
Fractional ideal class (41, 1/2*a + 5)
sage: J = G(I.ideal()^5); J
Fractional ideal class (115856201, 1/2*a + 40407883)
sage: J.reduce()
Fractional ideal class (57, 1/2*a + 44)
sage: J == I^5
True
representative_prime ( norm_bound=1000)
Return a prime ideal in this ideal class.
INPUT:
norm_bound (positive integer) – upper bound on the norm of primes tested.
EXAMPLES:
sage: K.<a> = NumberField(x^2+31)
sage: K.class_number()
3
sage: Cl = K.class_group()
sage: [c.representative_prime() for c in Cl]
[Fractional ideal (3),
Fractional ideal (2, 1/2*a + 1/2),
Fractional ideal (2, 1/2*a - 1/2)]
sage: K.<a> = NumberField(x^2+223)
sage: K.class_number()
7
sage: Cl = K.class_group()
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sage: [c.representative_prime() for c in Cl]
[Fractional ideal (3),
Fractional ideal (2, 1/2*a + 1/2),
Fractional ideal (17, 1/2*a + 7/2),
Fractional ideal (7, 1/2*a - 1/2),
Fractional ideal (7, 1/2*a + 1/2),
Fractional ideal (17, 1/2*a + 27/2),
Fractional ideal (2, 1/2*a - 1/2)]
class sage.rings.number_field.class_group. SClassGroup ( gens_orders, names, number_field, gens, S, proof=True)
Bases: sage.rings.number_field.class_group.ClassGroup
The S-class group of a number field.
EXAMPLES:
sage: K.<a> = QuadraticField(-14)
sage: S = K.primes_above(2)
sage: K.S_class_group(S).gens()
# random gens (platform dependent)
(Fractional S-ideal class (3, a + 2),)
sage: K.<a> = QuadraticField(-974)
sage: CS = K.S_class_group(K.primes_above(2)); CS
S-class group of order 18 with structure C6 x C3
of Number Field in a with defining polynomial x^2 + 974
sage: CS.gen(0) # random
Fractional S-ideal class (3, a + 2)
sage: CS.gen(1) # random
Fractional S-ideal class (31, a + 24)
Element
alias of SFractionalIdealClass
S()
Return the set (or rather tuple) of primes used to define this class group.
EXAMPLES:
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S);CS
S-class group of order 2 with structure C2 of Number Field in a with defining
˓→polynomial x^2 + 14
sage: T = tuple([])
sage: CT = K.S_class_group(T);CT
S-class group of order 4 with structure C4 of Number Field in a with defining
˓→polynomial x^2 + 14
sage: CS.S()
(Fractional ideal (2, a),)
sage: CT.S()
()
class sage.rings.number_field.class_group. SFractionalIdealClass ( parent, element,
ideal=None)
Bases: sage.rings.number_field.class_group.FractionalIdealClass
An S-fractional ideal class in a number field for a tuple of primes S.
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EXAMPLES:
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I)
Trivial S-ideal class
sage: CS(J)
Trivial S-ideal class
sage: CS(G)
Fractional S-ideal class (3, a + 1)
EXAMPLES:
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: J = K.ideal(7,a)
sage: G = K.ideal(3,a+1)
sage: CS(I).ideal()
Fractional ideal (2, a)
sage: CS(J).ideal()
Fractional ideal (7, a)
sage: CS(G).ideal()
Fractional ideal (3, a + 1)
EXAMPLES:
sage: K.<a> = QuadraticField(-14)
sage: I = K.ideal(2,a)
sage: S = (I,)
sage: CS = K.S_class_group(S)
sage: G = K.ideal(3,a+1)
sage: CS(G).inverse()
Fractional S-ideal class (3, a + 2)
3.5 Unit and S-unit groups of Number Fields
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4-8*x^2+36)
sage: UK = UnitGroup(K); UK
Unit group with structure C4 x Z of Number Field in a with defining polynomial x^4 ˓→8*x^2 + 36
The first generator is a primitive root of unity in the field:
sage: UK.gens()
(u0, u1)
sage: UK.gens_values() # random
[-1/12*a^3 + 1/6*a, 1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
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sage: UK.gen(0).value()
1/12*a^3 - 1/6*a
sage: UK.gen(0)
u0
sage: UK.gen(0) + K.one()
1/12*a^3 - 1/6*a + 1
# coerce abstract generator into number field
sage: [u.multiplicative_order() for u in UK.gens()]
[4, +Infinity]
sage: UK.rank()
1
sage: UK.ngens()
2
Units in the field can be converted into elements of the unit group represented as elements of an abstract multiplicative
group:
sage: UK(1)
1
sage: UK(-1)
u0^2
sage: [UK(u) for u in (x^4-1).roots(K,multiplicities=False)]
[1, u0^2, u0, u0^3]
sage: UK.fundamental_units() # random
[1/24*a^3 + 1/4*a^2 - 1/12*a - 1]
sage: torsion_gen = UK.torsion_generator(); torsion_gen
u0
sage: torsion_gen.value()
1/12*a^3 - 1/6*a
sage: UK.zeta_order()
4
sage: UK.roots_of_unity()
[1/12*a^3 - 1/6*a, -1, -1/12*a^3 + 1/6*a, 1]
Exp and log functions provide maps between units as field elements and exponent vectors with respect to the generators:
sage: u = UK.exp([13,10]); u # random
-41/8*a^3 - 55/4*a^2 + 41/4*a + 55
sage: UK.log(u)
(1, 10)
sage: u = UK.fundamental_units()[0]
sage: [UK.log(u^k) == (0,k) for k in range(10)]
[True, True, True, True, True, True, True, True, True, True]
sage: all([UK.log(u^k) == (0,k) for k in range(10)])
True
sage: K.<a> = NumberField(x^5-2,'a')
sage: UK = UnitGroup(K)
sage: UK.rank()
2
sage: UK.fundamental_units()
[a^3 + a^2 - 1, a - 1]
S-unit groups may be constructed, where S is a set of primes:
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sage: K.<a> = NumberField(x^6+2)
sage: S = K.ideal(3).prime_factors(); S
[Fractional ideal (3, a + 1), Fractional ideal (3, a - 1)]
sage: SUK = UnitGroup(K,S=tuple(S)); SUK
S-unit group with structure C2 x Z x Z x Z x Z of Number Field in a with defining
˓→polynomial x^6 + 2 with S = (Fractional ideal (3, a + 1), Fractional ideal (3, a ˓→1))
sage: SUK.primes()
(Fractional ideal (3, a + 1), Fractional ideal (3, a - 1))
sage: SUK.rank()
4
sage: SUK.gens_values()
[-1, a^2 + 1, a^5 + a^4 - a^2 - a - 1, a + 1, -a + 1]
sage: u = 9*prod(SUK.gens_values()); u
-18*a^5 - 18*a^4 - 18*a^3 - 9*a^2 + 9*a + 27
sage: SUK.log(u)
(1, 3, 1, 7, 7)
sage: u == SUK.exp((1,3,1,7,7))
True
A relative number field example:
sage: L.<a, b> = NumberField([x^2 + x + 1, x^4 + 1])
sage: UL = L.unit_group(); UL
Unit group with structure C24 x Z x Z x Z of Number Field in a with defining
˓→polynomial x^2 + x + 1 over its base field
sage: UL.gens_values() # random
[-b^3*a - b^3, -b^3*a + b, (-b^3 - b^2 - b)*a - b - 1, (-b^3 - 1)*a - b^2 + b - 1]
sage: UL.zeta_order()
24
sage: UL.roots_of_unity()
[-b*a,
-b^2*a - b^2,
-b^3,
-a,
-b*a - b,
-b^2,
b^3*a,
-a - 1,
-b,
b^2*a,
b^3*a + b^3,
-1,
b*a,
b^2*a + b^2,
b^3,
a,
b*a + b,
b^2,
-b^3*a,
a + 1,
b,
-b^2*a,
-b^3*a - b^3,
1]
A relative extension example, which worked thanks to the code review by F.W.Clarke:
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sage: PQ.<X> = QQ[]
sage: F.<a, b> = NumberField([X^2 - 2, X^2
sage: PF.<Y> = F[]
sage: K.<c> = F.extension(Y^2 - (1 + a)*(a
sage: K.unit_group()
Unit group with structure C2 x Z x Z x Z x
˓→defining polynomial Y^2 + (-2*b - 3)*a -
- 3])
+ b)*a*b)
Z x Z x Z x Z of Number Field in c with
2*b - 6 over its base field
AUTHOR:
• John Cremona
class sage.rings.number_field.unit_group. UnitGroup ( number_field,
proof=True,
S=None)
Bases: sage.groups.abelian_gps.values.AbelianGroupWithValues_class
The unit group or an S-unit group of a number field.
exp ( exponents)
Return unit with given exponents with respect to group generators.
INPUT:
•u – Any object from which an element of the unit group’s number field 𝐾 may be constructed; an
error is raised if an element of 𝐾 cannot be constructed from u, or if the element constructed is not a
unit.
OUTPUT: a list of integers giving the exponents of u with respect to the unit group’s basis.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<z> = CyclotomicField(13)
sage: UK = UnitGroup(K)
sage: [UK.log(u) for u in UK.gens()]
[(1, 0, 0, 0, 0, 0),
(0, 1, 0, 0, 0, 0),
(0, 0, 1, 0, 0, 0),
(0, 0, 0, 1, 0, 0),
(0, 0, 0, 0, 1, 0),
(0, 0, 0, 0, 0, 1)]
sage: vec = [65,6,7,8,9,10]
sage: unit = UK.exp(vec)
sage: UK.log(unit)
(13, 6, 7, 8, 9, 10)
sage: UK.exp(UK.log(u)) == u.value()
True
An S-unit example:
sage: SUK = UnitGroup(K,S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
-8732*z^11 + 15496*z^10 + 51840*z^9 + 68804*z^8 + 51840*z^7 + 15496*z^6 ˓→8732*z^5 + 34216*z^3 + 64312*z^2 + 64312*z + 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
True
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fundamental_units ( )
Return generators for the free part of the unit group, as a list.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 + 23)
sage: U = UnitGroup(K)
sage: U.fundamental_units() # random
[1/4*a^3 - 7/4*a^2 + 17/4*a - 19/4]
log ( u)
Return the exponents of the unit u with respect to group generators.
INPUT:
•u – Any object from which an element of the unit group’s number field 𝐾 may be constructed; an
error is raised if an element of 𝐾 cannot be constructed from u, or if the element constructed is not a
unit.
OUTPUT: a list of integers giving the exponents of u with respect to the unit group’s basis.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<z> = CyclotomicField(13)
sage: UK = UnitGroup(K)
sage: [UK.log(u) for u in UK.gens()]
[(1, 0, 0, 0, 0, 0),
(0, 1, 0, 0, 0, 0),
(0, 0, 1, 0, 0, 0),
(0, 0, 0, 1, 0, 0),
(0, 0, 0, 0, 1, 0),
(0, 0, 0, 0, 0, 1)]
sage: vec = [65,6,7,8,9,10]
sage: unit = UK.exp(vec); unit # random
-253576*z^11 + 7003*z^10 - 395532*z^9 - 35275*z^8 - 500326*z^7 - 35275*z^6 ˓→395532*z^5 + 7003*z^4 - 253576*z^3 - 59925*z - 59925
sage: UK.log(unit)
(13, 6, 7, 8, 9, 10)
An S-unit example:
sage: SUK = UnitGroup(K,S=2)
sage: v = (3,1,4,1,5,9,2)
sage: u = SUK.exp(v); u
-8732*z^11 + 15496*z^10 + 51840*z^9 + 68804*z^8 + 51840*z^7 + 15496*z^6 ˓→8732*z^5 + 34216*z^3 + 64312*z^2 + 64312*z + 34216
sage: SUK.log(u)
(3, 1, 4, 1, 5, 9, 2)
sage: SUK.log(u) == v
True
number_field ( )
Return the number field associated with this unit group.
EXAMPLES:
sage: U = UnitGroup(QuadraticField(-23, 'w')); U
Unit group with structure C2 of Number Field in w with defining polynomial x^
˓→2 + 23
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sage: U.number_field()
Number Field in w with defining polynomial x^2 + 23
primes ( )
Return the (possibly empty) list of primes associated with this S-unit group.
EXAMPLES:
sage: K.<a> = QuadraticField(-23)
sage: S = tuple(K.ideal(3).prime_factors()); S
(Fractional ideal (3, 1/2*a - 1/2), Fractional ideal (3, 1/2*a + 1/2))
sage: U = UnitGroup(K,S=tuple(S)); U
S-unit group with structure C2 x Z x Z of Number Field in a with defining
˓→polynomial x^2 + 23 with S = (Fractional ideal (3, 1/2*a - 1/2), Fractional
˓→ideal (3, 1/2*a + 1/2))
sage: U.primes() == S
True
rank ( )
Return the rank of the unit group.
EXAMPLES:
sage: K.<z> = CyclotomicField(13)
sage: UnitGroup(K).rank()
5
sage: SUK = UnitGroup(K,S=2); SUK.rank()
6
roots_of_unity ( )
Return all the roots of unity in this unit group, primitive or not.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<b> = NumberField(x^2+1)
sage: U = UnitGroup(K)
sage: zs = U.roots_of_unity(); zs
[b, -1, -b, 1]
sage: [ z**U.zeta_order() for z in zs ]
[1, 1, 1, 1]
torsion_generator ( )
Return a generator for the torsion part of the unit group.
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - x^2 + 4)
sage: U = UnitGroup(K)
sage: U.torsion_generator()
u0
sage: U.torsion_generator().value() # random
-1/4*a^3 - 1/4*a + 1/2
zeta ( n=2, all=False)
Return one, or a list of all, primitive n-th root of unity in this unit group.
EXAMPLES:
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sage: x = polygen(QQ)
sage: K.<z> = NumberField(x^2 + 3)
sage: U = UnitGroup(K)
sage: U.zeta(1)
1
sage: U.zeta(2)
-1
sage: U.zeta(2, all=True)
[-1]
sage: U.zeta(3)
-1/2*z - 1/2
sage: U.zeta(3, all=True)
[-1/2*z - 1/2, 1/2*z - 1/2]
sage: U.zeta(4)
Traceback (most recent call last):
...
ValueError: n (=4) does not divide order of generator
sage: r.<x> = QQ[]
sage: K.<b> = NumberField(x^2+1)
sage: U = UnitGroup(K)
sage: U.zeta(4)
b
sage: U.zeta(4,all=True)
[b, -b]
sage: U.zeta(3)
Traceback (most recent call last):
...
ValueError: n (=3) does not divide order of generator
sage: U.zeta(3,all=True)
[]
zeta_order ( )
Returns the order of the torsion part of the unit group.
EXAMPLES:
sage:
sage:
sage:
sage:
6
x = polygen(QQ)
K.<a> = NumberField(x^4 - x^2 + 4)
U = UnitGroup(K)
U.zeta_order()
3.6 Small primes of degree one
Iterator for finding several primes of absolute degree one of a number field of small prime norm.
Algorithm:
Let 𝑃 denote the product of some set of prime numbers. (In practice, we use the product of the first 10000 primes,
because Pari computes this many by default.)
Let 𝐾 be a number field and let 𝑓 (𝑥) be a polynomial defining 𝐾 over the rational field. Let 𝛼 be a root of 𝑓 in 𝐾.
We know that [𝑂𝐾 : Z[𝛼]]2 = |∆(𝑓 (𝑥))/∆(𝑂𝐾 )|, where ∆ denotes the discriminant (see, for example, Proposition
4.4.4, p165 of [C]). Therefore, after discarding primes dividing ∆(𝑓 (𝑥)) (this includes all ramified primes), any
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integer 𝑛 such that gcd(𝑓 (𝑛), 𝑃 ) > 0 yields a prime 𝑝|𝑃 such that 𝑓 (𝑥) has a root modulo 𝑝. By the condition on
discriminants, this root is a single root. As is well known (see, for example Theorem 4.8.13, p199 of [C]), the ideal
generated by (𝑝, 𝛼 − 𝑛) is prime and of degree one.
Warning: It is possible that there are no primes of 𝐾 of absolute degree one of small prime norm, and it is
possible that this algorithm will not find any primes of small norm.
To do:
There are situations when this will fail. There are questions of finding primes of relative degree one. There are
questions of finding primes of exact degree larger than one. In short, if you can contribute, please do!
EXAMPLES:
sage: x = ZZ['x'].gen()
sage: F.<a> = NumberField(x^2 - 2)
sage: Ps = F.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (2*a + 1), Fractional ideal (-3*a + 1), Fractional ideal (-a + 5)]
sage: [ P.norm() for P in Ps ] # random
[7, 17, 23]
sage: all(ZZ(P.norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
The next two examples are for relative number fields.:
sage: L.<b> = F.extension(x^3 - a)
sage: Ps = L.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (17, b - 5), Fractional ideal (23, b - 4), Fractional ideal (31, b ˓→ 2)]
sage: [ P.absolute_norm() for P in Ps ] # random
[17, 23, 31]
sage: all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
sage: M.<c> = NumberField(x^2 - x*b^2 + b)
sage: Ps = M.primes_of_degree_one_list(3)
sage: Ps # random
[Fractional ideal (17, c - 2), Fractional ideal (c - 1), Fractional ideal (41, c +
˓→15)]
sage: [ P.absolute_norm() for P in Ps ] # random
[17, 31, 41]
sage: all(ZZ(P.absolute_norm()).is_prime() for P in Ps)
True
sage: all(P.residue_class_degree() == 1 for P in Ps)
True
AUTHORS:
• Nick Alexander (2008)
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• David Loeffler (2009): fixed a bug with relative fields
class sage.rings.number_field.small_primes_of_degree_one. Small_primes_of_degree_one_iter ( field
num
max
Iterator that finds primes of a number field of absolute degree one and bounded small prime norm.
INPUT:
•field – a NumberField .
•num_integer_primes (default: 10000) – an integer. We try to find primes of absolute norm no
greater than the num_integer_primes -th prime number. For example, if num_integer_primes
is 2, the largest norm found will be 3, since the second prime is 3.
•max_iterations (default: 100) – an integer. We test max_iterations integers to find small
primes before raising StopIteration .
AUTHOR:
•Nick Alexander
next ( )
Return a prime of absolute degree one of small prime norm.
Raises StopIteration if such a prime cannot be easily found.
EXAMPLES:
sage: x = QQ['x'].gen()
sage: K.<a> = NumberField(x^2 - 3)
sage: it = K.primes_of_degree_one_iter()
sage: [ next(it) for i in range(3) ] # random
[Fractional ideal (2*a + 1), Fractional ideal (-a + 4), Fractional ideal (3*a
˓→+ 2)]
We test that trac ticket #6396 is fixed. Note that the doctest is flagged as random since the string representation of ideals is somewhat unpredictable:
sage: N.<a,b> = NumberField([x^2 + 1, x^2 - 5])
sage: ids = N.primes_of_degree_one_list(10); ids # random
[Fractional ideal ((-1/2*b + 1/2)*a + 2),
Fractional ideal (-b*a + 1/2*b + 1/2),
Fractional ideal ((1/2*b + 3/2)*a - b),
Fractional ideal ((-1/2*b - 3/2)*a + b - 1),
Fractional ideal (-b*a - b + 1),
Fractional ideal (3*a + 1/2*b - 1/2),
Fractional ideal ((-3/2*b + 1/2)*a + 1/2*b - 1/2),
Fractional ideal ((-1/2*b - 5/2)*a - b + 1),
Fractional ideal (2*a - 3/2*b - 1/2),
Fractional ideal (3*a + 1/2*b + 5/2)]
sage: [x.absolute_norm() for x in ids]
[29, 41, 61, 89, 101, 109, 149, 181, 229, 241]
sage: ids[9] == N.ideal(3*a + 1/2*b + 5/2)
True
See also:
Finite residue fields
3.6. Small primes of degree one
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CHAPTER
FOUR
TOTALLY REAL FIELDS
4.1 Enumeration of Primitive Totally Real Fields
This module contains functions for enumerating all primitive totally real number fields of given degree and small
discriminant. Here a number field is called primitive if it contains no proper subfields except Q.
See also sage.rings.number_field.totallyreal_rel , which handles the non-primitive case using relative extensions.
4.1.1 Algorithm
We use Hunter’s algorithm ([Cohen2000], Section 9.3) with modifications due to Takeuchi [Takeuchi1999] and the
author [Voight2008].
We enumerate polynomials 𝑓 (𝑥) = 𝑥𝑛 + 𝑎𝑛−1 𝑥𝑛−1 + · · · + 𝑎0 . Hunter’s theorem gives bounds on 𝑎𝑛−1 and 𝑎𝑛−2 ;
then given 𝑎𝑛−1 and 𝑎𝑛−2 , one can recursively compute bounds on 𝑎𝑛−3 , . . . , 𝑎0 , using the fact that the polynomial
is totally real by looking at the zeros of successive derivatives and applying Rolle’s theorem. See [Takeuchi1999] for
more details.
4.1.2 Examples
In this first simple example, we compute the totally real quadratic fields of discriminant ≤ 50.
sage: enumerate_totallyreal_fields_prim(2,50)
[[5, x^2 - x - 1],
[8, x^2 - 2],
[12, x^2 - 3],
[13, x^2 - x - 3],
[17, x^2 - x - 4],
[21, x^2 - x - 5],
[24, x^2 - 6],
[28, x^2 - 7],
[29, x^2 - x - 7],
[33, x^2 - x - 8],
[37, x^2 - x - 9],
[40, x^2 - 10],
[41, x^2 - x - 10],
[44, x^2 - 11]]
sage: [ d for d in range(5,50) if (is_squarefree(d) and d%4 == 1) or (d%4 == 0 and is_
˓→squarefree(d/4)) ]
[5, 8, 12, 13, 17, 20, 21, 24, 28, 29, 33, 37, 40, 41, 44]
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Next, we compute all totally real quintic fields of discriminant ≤ 105 :
sage: ls = enumerate_totallyreal_fields_prim(5,10^5) ; ls
[[14641, x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1],
[24217, x^5 - 5*x^3 - x^2 + 3*x + 1],
[36497, x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1],
[38569, x^5 - 5*x^3 + 4*x - 1],
[65657, x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1],
[70601, x^5 - x^4 - 5*x^3 + 2*x^2 + 3*x - 1],
[81509, x^5 - x^4 - 5*x^3 + 3*x^2 + 5*x - 2],
[81589, x^5 - 6*x^3 + 8*x - 1],
[89417, x^5 - 6*x^3 - x^2 + 8*x + 3]]
sage: len(ls)
9
We see that there are 9 such fields (up to isomorphism!).
4.1.3 References
4.1.4 Authors
• John Voight (2007-09-01): Initial version.
• John Voight (2007-09-19): Various optimization tweaks.
• John Voight (2007-10-09): Added DSage module.
• John Voight (2007-10-17): Added pari functions to avoid recomputations.
• John Voight (2007-10-27): Separated DSage component.
• Craig Citro and John Voight (2007-11-04): Additional doctests and type checking.
• Craig Citro and John Voight (2008-02-10): Final modifications for submission.
sage.rings.number_field.totallyreal. enumerate_totallyreal_fields_prim ( n,
B,
a=[],
verbose=0,
return_seqs=False,
phc=False,
keep_fields=False,
t_2=False,
just_print=False,
return_pari_objects=True)
This function enumerates primitive totally real fields of degree 𝑛 > 1 with discriminant 𝑑 ≤ 𝐵; optionally one
can specify the first few coefficients, where the sequence 𝑎 corresponds to
a[d]*x^n + ... + a[0]*x^(n-d)
where length(a) = d+1 , so in particular always a[d] = 1 .
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Note: This is guaranteed to give all primitive such fields, and seems in practice to give many imprimitive ones.
INPUT:
•n – (integer) the degree
•B – (integer) the discriminant bound
•a – (list, default: []) the coefficient list to begin with
•verbose – (integer or string, default: 0) if verbose == 1 (or 2 ), then print to the screen (really)
verbosely; if verbose is a string, then print verbosely to the file specified by verbose.
•return_seqs – (boolean, default False) If True , then return the polynomials as sequences (for easier
exporting to a file).
•phc – boolean or integer (default: False)
•keep_fields – (boolean or integer, default: False) If keep_fields is True, then keep fields up to
B*log(B) ; if keep_fields is an integer, then keep fields up to that integer.
•t_2 – (boolean or integer, default: False) If t_2 = T , then keep only polynomials with t_2 norm >= T.
•just_print – (boolean, default: False): if just_print is not False, instead of creating a sorted list
of totally real number fields, we simply write each totally real field we find to the file whose filename is
given by just_print . In this case, we don’t return anything.
•return_pari_objects
– (boolean, default:
True) if both return_seqs
and
return_pari_objects are False then it returns the elements as Sage objects; otherwise it
returns pari objects.
OUTPUT:
the list of fields with entries [d,f] , where d is the discriminant and f is a defining polynomial, sorted by
discriminant.
AUTHORS:
•John Voight (2007-09-03)
•Craig Citro (2008-09-19): moved to Cython for speed improvement
sage.rings.number_field.totallyreal. odlyzko_bound_totallyreal ( n)
This function returns the unconditional Odlyzko bound for the root discriminant of a totally real number field of
degree n.
Note: The bounds for n > 50 are not necessarily optimal.
INPUT:
•n (integer) the degree
OUTPUT:
a lower bound on the root discriminant (as a real number)
EXAMPLES:
sage: [sage.rings.number_field.totallyreal.odlyzko_bound_totallyreal(n) for n in
˓→range(1,5)]
[1.0, 2.223, 3.61, 5.067]
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AUTHORS:
•John Voight (2007-09-03)
NOTES: The values are calculated by Martinet [Martinet1980].
sage.rings.number_field.totallyreal. timestr ( m)
Converts seconds to a human-readable time string.
INPUT:
•m – integer, number of seconds
OUTPUT:
The time in days, hours, etc.
EXAMPLES:
sage: sage.rings.number_field.totallyreal.timestr(3765)
'1h 2m 45.0s'
sage.rings.number_field.totallyreal. weed_fields ( S, lenS=0)
Function used internally by the enumerate_totallyreal_fields_prim() routine. (Weeds the fields
listed by [discriminant, polynomial] for isomorphism classes.) Returns the size of the resulting list.
EXAMPLES:
sage: ls = [[5,pari('x^2-3*x+1')],[5,pari('x^2-5')]]
sage: sage.rings.number_field.totallyreal.weed_fields(ls)
1
sage: ls
[[5, x^2 - 3*x + 1]]
4.2 Enumeration of Totally Real Fields: Relative Extensions
This module contains functions to enumerate primitive extensions 𝐿/𝐾, where 𝐾 is a given totally real number field, with given degree and small root discriminant. This is a relative analogue of the problem described in
sage.rings.number_field.totallyreal , and we use a similar approach based on a relative version of
Hunter’s theorem.
√
In this first simple example, we compute the totally real quadratic fields of 𝐹 = Q( 2) of discriminant ≤ 2000.
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
√
There is indeed only one such extension, given by 𝐹 ( 5).
√
Next, we list all totally real quadratic extensions of Q( 5) with root discriminant ≤ 10.
sage: F.<t> = NumberField(x^2-5)
sage: ls = enumerate_totallyreal_fields_rel(F, 2, 10^4)
sage: ls # random (the second factor is platform-dependent)
[[725, x^4 - x^3 - 3*x^2 + x + 1, xF^2 + (-1/2*t - 7/2)*xF + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1, xF^2 + (-1/2*t - 7/2)*xF + 1/2*t + 3/2],
[1600, x^4 - 6*x^2 + 4, xF^2 - 2],
[2000, x^4 - 5*x^2 + 5, xF^2 - 1/2*t - 5/2],
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[2225, x^4 - x^3 - 5*x^2 + 2*x + 4, xF^2 + (-1/2*t + 1/2)*xF - 3/2*t - 7/2],
[2525, x^4 - 2*x^3 - 4*x^2 + 5*x + 5, xF^2 + (-1/2*t - 1/2)*xF - 1/2*t - 5/2],
[3600, x^4 - 2*x^3 - 7*x^2 + 8*x + 1, xF^2 - 3],
[4225, x^4 - 9*x^2 + 4, xF^2 + (-1/2*t - 1/2)*xF - 3/2*t - 9/2],
[4400, x^4 - 7*x^2 + 11, xF^2 - 1/2*t - 7/2],
[4525, x^4 - x^3 - 7*x^2 + 3*x + 9, xF^2 + (-1/2*t - 1/2)*xF - 3],
[5125, x^4 - 2*x^3 - 6*x^2 + 7*x + 11, xF^2 + (-1/2*t - 1/2)*xF - t - 4],
[5225, x^4 - x^3 - 8*x^2 + x + 11, xF^2 + (-1/2*t - 1/2)*xF - 1/2*t - 7/2],
[5725, x^4 - x^3 - 8*x^2 + 6*x + 11, xF^2 + (-1/2*t + 1/2)*xF - 1/2*t - 7/2],
[6125, x^4 - x^3 - 9*x^2 + 9*x + 11, xF^2 + (-1/2*t + 1/2)*xF - t - 4],
[7225, x^4 - 11*x^2 + 9, xF^2 + (-1)*xF - 4],
[7600, x^4 - 9*x^2 + 19, xF^2 - 1/2*t - 9/2],
[7625, x^4 - x^3 - 9*x^2 + 4*x + 16, xF^2 + (-1/2*t - 1/2)*xF - 4],
[8000, x^4 - 10*x^2 + 20, xF^2 - t - 5],
[8525, x^4 - 2*x^3 - 8*x^2 + 9*x + 19, xF^2 + (-1)*xF - 1/2*t - 9/2],
[8725, x^4 - x^3 - 10*x^2 + 2*x + 19, xF^2 + (-1/2*t - 1/2)*xF - 1/2*t - 9/2],
[9225, x^4 - x^3 - 10*x^2 + 7*x + 19, xF^2 + (-1/2*t + 1/2)*xF - 1/2*t - 9/2]]
sage: [ f[0] for f in ls ]
[725, 1125, 1600, 2000, 2225, 2525, 3600, 4225, 4400, 4525, 5125, 5225, 5725, 6125,
˓→7225, 7600, 7625, 8000, 8525, 8725, 9225]
sage: [NumberField(ZZx(x[1]), 't').is_galois() for x in ls]
[False, True, True, True, False, False, True, True, False, False, False, False,
˓→False, True, True, False, False, True, False, False, False]
Eight out of 21 such fields are Galois (with Galois group 𝐶4 or 𝐶2 ×𝐶2 ); the others have have Galois closure of degree
8 (with Galois group 𝐷8 ).
Finally, we compute the cubic extensions of Q(𝜁7 )+ with discriminant ≤ 17 × 109 .
sage: F.<t> = NumberField(ZZx([1,-4,3,1]))
sage: F.disc()
49
sage: enumerate_totallyreal_fields_rel(F, 3, 17*10^9) # not tested, too long time
˓→(258s on sage.math, 2013)
[[16240385609L, x^9 - x^8 - 9*x^7 + 4*x^6 + 26*x^5 - 2*x^4 - 25*x^3 - x^2 + 7*x + 1,
˓→xF^3 + (-t^2 - 4*t + 1)*xF^2 + (t^2 + 3*t - 5)*xF + 3*t^2 + 11*t - 5]]
# 32-bit
[[16240385609, x^9 - x^8 - 9*x^7 + 4*x^6 + 26*x^5 - 2*x^4 - 25*x^3 - x^2 + 7*x + 1,
˓→xF^3 + (-t^2 - 4*t + 1)*xF^2 + (t^2 + 3*t - 5)*xF + 3*t^2 + 11*t - 5]]
# 64-bit
AUTHORS:
• John Voight (2007-11-03): Initial version.
sage.rings.number_field.totallyreal_rel. enumerate_totallyreal_fields_all ( n,
B,
verbose=0,
return_seqs=False,
return_pari_objects=True
Enumerates all totally real fields of degree n with discriminant at most B , primitive or otherwise.
INPUT:
•n – integer, the degree
•B – integer, the discriminant bound
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•verbose – boolean or nonnegative integer or string (default: 0) give a verbose description of the computations being performed. If verbose is set to 2 or more then it outputs some extra information. If
verbose is a string then it outputs to a file specified by verbose
•return_seqs – (boolean, default False) If True , then return the polynomials as sequences (for easier
exporting to a file). This also returns a list of four numbers, as explained in the OUTPUT section below.
•return_pari_objects
– (boolean, default:
True) if both return_seqs
and
return_pari_objects are False then it returns the elements as Sage objects; otherwise it
returns pari objects.
EXAMPLES:
sage: enumerate_totallyreal_fields_all(4, 2000)
[[725, x^4 - x^3 - 3*x^2 + x + 1],
[1125, x^4 - x^3 - 4*x^2 + 4*x + 1],
[1600, x^4 - 6*x^2 + 4],
[1957, x^4 - 4*x^2 - x + 1],
[2000, x^4 - 5*x^2 + 5]]
sage: enumerate_totallyreal_fields_all(1, 10)
[[1, x - 1]]
sage.rings.number_field.totallyreal_rel. enumerate_totallyreal_fields_rel ( F,
m,
B,
a=[],
verbose=0,
return_seqs=False,
return_pari_objects=True
This function enumerates (primitive) totally real field extensions of degree 𝑚 > 1 of the totally real field F with
discriminant 𝑑 ≤ 𝐵; optionally one can specify the first few coefficients, where the sequence a corresponds to
a polynomial by
a[d]*x^n + ... + a[0]*x^(n-d)
if length(a) = d+1 , so in particular always a[d] = 1 .
Note: This is guaranteed to give all primitive such fields, and seems in practice to give many imprimitive ones.
INPUT:
•F – number field, the base field
•m – integer, the degree
•B – integer, the discriminant bound
•a – list (default: []), the coefficient list to begin with
•verbose – boolean or nonnegative integer or string (default: 0) give a verbose description of the computations being performed. If verbose is set to 2 or more then it outputs some extra information. If
verbose is a string then it outputs to a file specified by verbose
•return_seqs – (boolean, default False) If True , then return the polynomials as sequences (for easier
exporting to a file). This also returns a list of four numbers, as explained in the OUTPUT section below.
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•return_pari_objects
– (boolean, default:
True) if both return_seqs
and
return_pari_objects are False then it returns the elements as Sage objects; otherwise it
returns pari objects.
OUTPUT:
•the list of fields with entries [d,fabs,f] , where d is the discriminant, fabs is an absolute defining
polynomial, and f is a defining polynomial relative to F , sorted by discriminant.
•if return_seqs is True , then the first field of the list is a list containing the count of four items as
explained below
–the first entry gives the number of polynomials tested
–the second entry gives the number of polynomials with its discriminant having a large enough square
divisor
–the third entry is the number of irreducible polynomials
–the fourth entry is the number of irreducible polynomials with discriminant at most B
EXAMPLES:
sage: ZZx = ZZ['x']
sage: F.<t> = NumberField(x^2-2)
sage: enumerate_totallyreal_fields_rel(F, 1, 2000)
[[1, [-2, 0, 1], xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000)
[[1600, x^4 - 6*x^2 + 4, xF^2 + xF - 1]]
sage: enumerate_totallyreal_fields_rel(F, 2, 2000, return_seqs=True)
[[9, 6, 5, 0], [[1600, [4, 0, -6, 0, 1], [-1, 1, 1]]]]
AUTHORS:
•John Voight (2007-11-01)
sage.rings.number_field.totallyreal_rel. integral_elements_in_box ( K, C)
Return all integral elements of the totally real field 𝐾 whose embeddings lie numerically within the bounds
specified by the list 𝐶. The output is architecture dependent, and one may want to expand the bounds that define
C by some epsilon.
INPUT:
•𝐾 – a totally real number field
•𝐶 – a list [[lower, upper], ...] of lower and upper bounds, for each embedding
EXAMPLES:
sage: x = polygen(QQ)
sage: K.<alpha> = NumberField(x^2-2)
sage: eps = 10e-6
sage: C = [[0-eps,5+eps],[0-eps,10+eps]]
sage: ls = sage.rings.number_field.totallyreal_rel.integral_elements_in_box(K, C)
sage: sorted([ a.trace() for a in ls ])
[0, 2, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 10, 10, 10, 10, 12, 12, 14]
sage: len(ls)
19
sage: v = sage.rings.number_field.totallyreal_rel.integral_elements_in_box(K, C)
sage: sorted(v)
[0, -alpha + 2, 1, -alpha + 3, 2, 3, alpha + 2, 4, alpha + 3, 5, alpha + 4,
˓→2*alpha + 3, alpha + 5, 2*alpha + 4, alpha + 6, 2*alpha + 5, 2*alpha + 6,
˓→3*alpha + 5, 2*alpha + 7]
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A cubic field:
sage:
sage:
sage:
sage:
sage:
x = polygen(QQ)
K.<a> = NumberField(x^3 - 16*x +16)
eps = 10e-6
C = [[0-eps,5+eps]]*3
v = sage.rings.number_field.totallyreal_rel.integral_elements_in_box(K, C)
Note that the output is platform dependent (sometimes a 5 is listed below, and sometimes it isn’t):
sage: sorted(v)
[-1/2*a + 2, 1/4*a^2 + 1/2*a, 0, 1, 2, 3, 4,...-1/4*a^2 - 1/2*a + 5, 1/2*a + 3, ˓→1/4*a^2 + 5]
class sage.rings.number_field.totallyreal_rel. tr_data_rel ( F, m, B, a=None)
This class encodes the data used in the enumeration of totally real fields for relative extensions.
We do not give a complete description here. For more information, see the attached functions; all of these are
used internally by the functions in totallyreal_rel.py, so see that file for examples and further documentation.
incr ( f_out, verbose=False, haltk=0)
This function ‘increments’ the totally real data to the next value which satisfies the bounds essentially
given by Rolle’s theorem, and returns the next polynomial in the sequence f_out.
The default or usual case just increments the constant coefficient; then inductively, if this is outside of the
bounds we increment the next higher coefficient, and so on.
If there are no more coefficients to be had, returns the zero polynomial.
INPUT:
•f_out – an integer sequence, to be written with the coefficients of the next polynomial
•verbose – boolean or nonnegative integer (default: False) print verbosely computational details. It
prints extra information if verbose is set to 2 or more
•haltk – integer, the level at which to halt the inductive coefficient bounds
OUTPUT:
the successor polynomial as a coefficient list.
4.3 Enumeration of Totally Real Fields
AUTHORS:
• Craig Citro and John Voight (2007-11-04): Type checking and other polishing.
• John Voight (2007-10-09): Improvements: Smyth bound, Lagrange multipliers for b.
• John Voight (2007-09-19): Various optimization tweaks.
• John Voight (2007-09-01): Initial version.
sage.rings.number_field.totallyreal_data. easy_is_irreducible_py ( f )
Used solely for testing easy_is_irreducible.
EXAMPLES:
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sage: sage.rings.number_field.totallyreal_data.easy_is_irreducible_py(pari('x^2+1
˓→'))
1
sage: sage.rings.number_field.totallyreal_data.easy_is_irreducible_py(pari('x^2-1
˓→'))
0
sage.rings.number_field.totallyreal_data. hermite_constant ( n)
This function returns the nth Hermite constant
The nth Hermite constant (typically denoted 𝛾𝑛 ), is defined to be
max min ||𝑥||2
𝐿
0̸=𝑥∈𝐿
where 𝐿 runs over all lattices of dimension 𝑛 and determinant 1.
For 𝑛 ≤ 8 it returns the exact value of 𝛾𝑛 , and for 𝑛 > 9 it returns an upper bound on 𝛾𝑛 .
INPUT:
•n – integer
OUTPUT:
•(an upper bound for) the Hermite constant gamma_n
EXAMPLES:
sage: hermite_constant(1) # trivial one-dimensional lattice
1.0
sage: hermite_constant(2) # Eisenstein lattice
1.1547005383792515
sage: 2/sqrt(3.)
1.15470053837925
sage: hermite_constant(8) # E_8
2.0
Note: The upper bounds used can be found in [CS] and [CE].
REFERENCES:
AUTHORS:
•John Voight (2007-09-03)
sage.rings.number_field.totallyreal_data. int_has_small_square_divisor ( d)
Returns the largest a such that a^2 divides d and a has prime divisors < 200.
EXAMPLES:
sage: from sage.rings.number_field.totallyreal_data import int_has_small_square_
˓→divisor
sage: int_has_small_square_divisor(500)
100
sage: is_prime(691)
True
sage: int_has_small_square_divisor(691)
1
sage: int_has_small_square_divisor(691^2)
1
4.3. Enumeration of Totally Real Fields
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sage.rings.number_field.totallyreal_data. lagrange_degree_3 ( n, an1, an2, an3)
Private function. Solves the equations which arise in the Lagrange multiplier for degree 3: for each 1 <= r <=
n-2, we solve
r*x^i + (n-1-r)*y^i + z^i = s_i (i = 1,2,3)
where the s_i are the power sums determined by the coefficients a. We output the largest value of z which
occurs. We use a precomputed elimination ideal.
EXAMPLES:
sage: ls = sage.rings.number_field.totallyreal_data.lagrange_degree_3(3,0,1,2)
sage: [RealField(10)(x) for x in ls]
[-1.0, -1.0]
sage: sage.rings.number_field.totallyreal_data.lagrange_degree_3(3,6,1,2) # random
[-5.8878, -5.8878]
class sage.rings.number_field.totallyreal_data. tr_data
Bases: object
This class encodes the data used in the enumeration of totally real fields.
We do not give a complete description here. For more information, see the attached functions; all of these are
used internally by the functions in totallyreal.py, so see that file for examples and further documentation.
increment ( verbose=False, haltk=0, phc=False)
This function ‘increments’ the totally real data to the next value which satisfies the bounds essentially
given by Rolle’s theorem, and returns the next polynomial as a sequence of integers.
The default or usual case just increments the constant coefficient; then inductively, if this is outside of the
bounds we increment the next higher coefficient, and so on.
If there are no more coefficients to be had, returns the zero polynomial.
INPUT:
•verbose – boolean to print verbosely computational details
•haltk – integer, the level at which to halt the inductive coefficient bounds
•phc – boolean, if PHCPACK is available, use it when k == n-5 to compute an improved Lagrange
multiplier bound
OUTPUT:
The next polynomial, as a sequence of integers
EXAMPLES:
sage: T = sage.rings.number_field.totallyreal_data.tr_data(2,100)
sage: T.increment()
[-24, -1, 1]
sage: for i in range(19): _ = T.increment()
sage: T.increment()
[-3, -1, 1]
sage: T.increment()
[-25, 0, 1]
printa ( )
Print relevant data for self.
EXAMPLES:
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sage: T = sage.rings.number_field.totallyreal_data.tr_data(3,2^10)
sage: T.printa()
k = 1
a = [0, 0, -1, 1]
amax = [0, 0, 0, 1]
beta = [...]
gnk = [...]
4.4 Enumeration of Totally Real Fields: PHC interface
AUTHORS:
– John Voight (2007-10-10):
• Zeroth attempt.
sage.rings.number_field.totallyreal_phc. coefficients_to_power_sums ( n, m, a)
Takes the list a, representing a list of initial coefficients of a (monic) polynomial of degree n, and returns the
power sums of the roots of f up to (m-1)th powers.
INPUT:
•n – integer, the degree
•a – list of integers, the coefficients
OUTPUT:
list of integers.
NOTES:
Uses Newton’s relations, which are classical.
AUTHORS:
•John Voight (2007-09-19)
EXAMPLES:
sage: sage.rings.number_field.totallyreal_phc.coefficients_to_power_
˓→sums(3,2,[1,5,7])
[3, -7, 39]
sage: sage.rings.number_field.totallyreal_phc.coefficients_to_power_
˓→sums(5,4,[1,5,7,9,8])
[5, -8, 46, -317, 2158]
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CHAPTER
FIVE
ALGEBRAIC NUMBERS
5.1 Field of Algebraic Numbers
AUTHOR:
• Carl Witty (2007-01-27): initial version
• Carl Witty (2007-10-29): massive rewrite to support complex as well as real numbers
This is an implementation of the algebraic numbers (the complex numbers which are the zero of a polynomial in Z[𝑥];
in other words, the algebraic closure of Q, with an embedding into C). All computations are exact. We also include
an implementation of the algebraic reals (the intersection of the algebraic numbers with R). The field of algebraic
numbers Q is available with abbreviation QQbar ; the field of algebraic reals has abbreviation AA .
As with many other implementations of the algebraic numbers, we try hard to avoid computing a number field and
working in the number field; instead, we use floating-point interval arithmetic whenever possible (basically whenever
we need to prove non-equalities), and resort to symbolic computation only as needed (basically to prove equalities).
Algebraic numbers exist in one of the following forms:
• a rational number
• the sum, difference, product, or quotient of algebraic numbers
• the negation, inverse, absolute value, norm, real part, imaginary part, or complex conjugate of an algebraic
number
• a particular root of a polynomial, given as a polynomial with algebraic coefficients together with an isolating
interval (given as a RealIntervalFieldElement ) which encloses exactly one root, and the multiplicity
of the root
• a polynomial in one generator, where the generator is an algebraic number given as the root of an irreducible
polynomial with integral coefficients and the polynomial is given as a NumberFieldElement .
An algebraic number can be coerced into ComplexIntervalField (or RealIntervalField , for algebraic
reals); every algebraic number has a cached interval of the highest precision yet calculated.
In most cases, computations that need to compare two algebraic numbers compute them with 128-bit precision intervals; if this does not suffice to prove that the numbers are different, then we fall back on exact computation.
Note that division involves an implicit comparison of the divisor against zero, and may thus trigger exact computation.
Also, using an algebraic number in the leading coefficient of a polynomial also involves an implicit comparison against
zero, which again may trigger exact computation.
Note that we work fairly hard to avoid computing new number fields; to help, we keep a lattice of already-computed
number fields and their inclusions.
EXAMPLES:
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sage: sqrt(AA(2)) > 0
True
sage: (sqrt(5 + 2*sqrt(QQbar(6))) - sqrt(QQbar(3)))^2 == 2
True
sage: AA((sqrt(5 + 2*sqrt(6)) - sqrt(3))^2) == 2
True
For a monic cubic polynomial 𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 with roots 𝑠1, 𝑠2, 𝑠3, the discriminant is defined as (𝑠1 − 𝑠2)2 (𝑠1 −
𝑠3)2 (𝑠2 − 𝑠3)2 and can be computed as 𝑏2 𝑐2 − 4𝑏3 𝑑 − 4𝑐3 + 18𝑏𝑐𝑑 − 27𝑑2 . We can test that these definitions do give
the same result:
sage:
....:
sage:
....:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
True
sage:
sage:
sage:
sage:
sage:
sage:
sage:
True
sage:
sage:
sage:
sage:
sage:
sage:
sage:
True
def disc1(b, c, d):
return b^2*c^2 - 4*b^3*d - 4*c^3 + 18*b*c*d - 27*d^2
def disc2(s1, s2, s3):
return ((s1-s2)*(s1-s3)*(s2-s3))^2
x = polygen(AA)
p = x*(x-2)*(x-4)
cp = AA.common_polynomial(p)
d, c, b, _ = p.list()
s1 = AA.polynomial_root(cp, RIF(-1, 1))
s2 = AA.polynomial_root(cp, RIF(1, 3))
s3 = AA.polynomial_root(cp, RIF(3, 5))
disc1(b, c, d) == disc2(s1, s2, s3)
p = p + 1
cp = AA.common_polynomial(p)
d, c, b, _ = p.list()
s1 = AA.polynomial_root(cp, RIF(-1, 1))
s2 = AA.polynomial_root(cp, RIF(1, 3))
s3 = AA.polynomial_root(cp, RIF(3, 5))
disc1(b, c, d) == disc2(s1, s2, s3)
p = (x-sqrt(AA(2)))*(x-AA(2).nth_root(3))*(x-sqrt(AA(3)))
cp = AA.common_polynomial(p)
d, c, b, _ = p.list()
s1 = AA.polynomial_root(cp, RIF(1.4, 1.5))
s2 = AA.polynomial_root(cp, RIF(1.7, 1.8))
s3 = AA.polynomial_root(cp, RIF(1.2, 1.3))
disc1(b, c, d) == disc2(s1, s2, s3)
We can convert from symbolic expressions:
sage: QQbar(sqrt(-5))
2.236067977499790?*I
sage: AA(sqrt(2) + sqrt(3))
3.146264369941973?
sage: QQbar(I)
I
sage: AA(I)
Traceback (most recent call last):
...
ValueError: Cannot coerce algebraic number with non-zero imaginary part to algebraic
˓→real
sage: QQbar(I * golden_ratio)
1.618033988749895?*I
sage: AA(golden_ratio)^2 - AA(golden_ratio)
1
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sage: QQbar((-8)^(1/3))
1.000000000000000? + 1.732050807568878?*I
sage: AA((-8)^(1/3))
-2
sage: QQbar((-4)^(1/4))
1 + 1*I
sage: AA((-4)^(1/4))
Traceback (most recent call last):
...
ValueError: Cannot coerce algebraic number with non-zero imaginary part to algebraic
˓→real
The coercion, however, goes in the other direction, since not all symbolic expressions are algebraic numbers:
sage: QQbar(sqrt(2)) + sqrt(3)
sqrt(3) + 1.414213562373095?
sage: QQbar(sqrt(2) + QQbar(sqrt(3)))
3.146264369941973?
Note the different behavior in taking roots: for AA we prefer real roots if they exist, but for QQbar we take the
principal root:
sage: AA(-1)^(1/3)
-1
sage: QQbar(-1)^(1/3)
0.500000000000000? + 0.866025403784439?*I
We can explicitly coerce from Q[𝐼]. (Technically, this is not quite kosher, since Q[𝐼] doesn’t come with an embedding;
we do not know whether the field generator is supposed to map to +𝐼 or −𝐼. We assume that for any quadratic field
with polynomial 𝑥2 + 1, the generator maps to +𝐼.):
sage:
sage:
4/5*I
sage:
True
K.<im> = QQ[I]
pythag = QQbar(3/5 + 4*im/5); pythag
+ 3/5
pythag.abs() == 1
However, implicit coercion from Q[𝐼] is not allowed:
sage: QQbar(1) + im
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Algebraic Field' and 'Number Field
˓→in I with defining polynomial x^2 + 1'
We can implicitly coerce from algebraic reals to algebraic numbers:
sage: a = QQbar(1); a, a.parent()
(1, Algebraic Field)
sage: b = AA(1); b, b.parent()
(1, Algebraic Real Field)
sage: c = a + b; c, c.parent()
(2, Algebraic Field)
Some computation with radicals:
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sage:
sage:
True
sage:
sage:
True
sage:
True
sage:
True
phi = (1 + sqrt(AA(5))) / 2
phi^2 == phi + 1
sage:
sage:
sage:
sage:
True
rt23
rt35
rt25
rt23
tau = (1 - sqrt(AA(5))) / 2
tau^2 == tau + 1
phi + tau == 1
tau < 0
=
=
=
*
sqrt(AA(2/3))
sqrt(AA(3/5))
sqrt(AA(2/5))
rt35 == rt25
The Sage rings AA and QQbar can decide equalities between radical expressions (over the reals and complex numbers
respectively):
sage: a = AA((2/(3*sqrt(3)) + 10/27)^(1/3) - 2/(9*(2/(3*sqrt(3)) + 10/27)^(1/3)) + 1/
˓→3)
sage: a
1.000000000000000?
sage: a == 1
True
Algebraic numbers which are known to be rational print as rationals; otherwise they print as intervals (with 53-bit
precision):
sage: AA(2)/3
2/3
sage: QQbar(5/7)
5/7
sage: QQbar(1/3 - 1/4*I)
-1/4*I + 1/3
sage: two = QQbar(4).nth_root(4)^2; two
2.000000000000000?
sage: two == 2; two
True
2
sage: phi
1.618033988749895?
We can find the real and imaginary parts of an algebraic number (exactly):
sage: r = QQbar.polynomial_root(x^5 - x - 1, CIF(RIF(0.1, 0.2), RIF(1.0, 1.1))); r
0.1812324444698754? + 1.083954101317711?*I
sage: r.real()
0.1812324444698754?
sage: r.imag()
1.083954101317711?
sage: r.minpoly()
x^5 - x - 1
sage: r.real().minpoly()
x^10 + 3/16*x^6 + 11/32*x^5 - 1/64*x^2 + 1/128*x - 1/1024
sage: r.imag().minpoly() # long time (10s on sage.math, 2013)
x^20 - 5/8*x^16 - 95/256*x^12 - 625/1024*x^10 - 5/512*x^8 - 1875/8192*x^6 + 25/4096*x^
˓→4 - 625/32768*x^2 + 2869/1048576
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We can find the absolute value and norm of an algebraic number exactly. (Note that we define the norm as the product
of a number and its complex conjugate; this is the algebraic definition of norm, if we view QQbar as AA[I] .):
sage: R.<x> = QQ[]
sage: r = (x^3 + 8).roots(QQbar, multiplicities=False)[2]; r
1.000000000000000? + 1.732050807568878?*I
sage: r.abs() == 2
True
sage: r.norm() == 4
True
sage: (r+QQbar(I)).norm().minpoly()
x^2 - 10*x + 13
sage: r = AA.polynomial_root(x^2 - x - 1, RIF(-1, 0)); r
-0.618033988749895?
sage: r.abs().minpoly()
x^2 + x - 1
We can compute the multiplicative order of an algebraic number:
sage: QQbar(-1/2 + I*sqrt(3)/2).multiplicative_order()
3
sage: QQbar(-sqrt(3)/2 + I/2).multiplicative_order()
12
sage: (QQbar.zeta(23)**5).multiplicative_order()
23
The paper “ARPREC: An Arbitrary Precision Computation Package” by Bailey, Yozo, Li and Thompson discusses
this result. Evidently it is difficult to find, but we can easily verify it.
sage: alpha = QQbar.polynomial_root(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1,
˓→RIF(1, 1.2))
sage: lhs = alpha^630 - 1
sage: rhs_num = (alpha^315 - 1) * (alpha^210 - 1) * (alpha^126 - 1)^2 * (alpha^90 ˓→1) * (alpha^3 - 1)^3 * (alpha^2 - 1)^5 * (alpha - 1)^3
sage: rhs_den = (alpha^35 - 1) * (alpha^15 - 1)^2 * (alpha^14 - 1)^2 * (alpha^5 - 1)^
˓→6 * alpha^68
sage: rhs = rhs_num / rhs_den
sage: lhs
2.642040335819351?e44
sage: rhs
2.642040335819351?e44
sage: lhs - rhs
0.?e29
sage: lhs == rhs
True
sage: lhs - rhs
0
sage: lhs._exact_value()
10648699402510886229334132989629606002223831*a^9 +
˓→23174560249100286133718183712802529035435800*a^8 +
˓→27259790692625442252605558473646959458901265*a^7 +
˓→21416469499004652376912957054411004410158065*a^6 +
˓→14543082864016871805545108986578337637140321*a^5 +
˓→6458050008796664339372667222902512216589785*a^4 ˓→3052219053800078449122081871454923124998263*a^3 ˓→14238966128623353681821644902045640915516176*a^2 ˓→16749022728952328254673732618939204392161001*a ˓→9052854758155114957837247156588012516273410 where a^10 + a^9 - a^7 - a^6 - a^5 - a^
˓→4 - a^3 + a + 1 = 0 and a in 1.176280818259918?
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Given an algebraic number, we can produce a string that will reproduce that algebraic number if you type the string
into Sage. We can see that until exact computation is triggered, an algebraic number keeps track of the computation
steps used to produce that number:
sage: rt2 = AA(sqrt(2))
sage: rt3 = AA(sqrt(3))
sage: n = (rt2 + rt3)^5; n
308.3018001722975?
sage: sage_input(n)
R.<x> = AA[]
v1 = AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.4142135623730949),
˓→RR(1.4142135623730951))) + AA.polynomial_root(AA.common_polynomial(x^2 - 3),
˓→RIF(RR(1.7320508075688772), RR(1.7320508075688774)))
v2 = v1*v1
v2*v2*v1
But once exact computation is triggered, the computation tree is discarded, and we get a way to produce the number
directly:
sage: n == 109*rt2 + 89*rt3
True
sage: sage_input(n)
R.<x> = AA[]
v = AA.polynomial_root(AA.common_polynomial(x^4 - 4*x^2 + 1), RIF(RR(0.
˓→51763809020504148), RR(0.51763809020504159)))
-109*v^3 - 89*v^2 + 327*v + 178
We can also see that some computations (basically, those which are easy to perform exactly) are performed directly,
instead of storing the computation tree:
sage: z3_3 = QQbar.zeta(3) * 3
sage: z4_4 = QQbar.zeta(4) * 4
sage: z5_5 = QQbar.zeta(5) * 5
sage: sage_input(z3_3 * z4_4 * z5_5)
R.<x> = AA[]
3*QQbar.polynomial_root(AA.common_polynomial(x^2 + x + 1), CIF(RIF(-RR(0.
˓→50000000000000011), -RR(0.49999999999999994)), RIF(RR(0.8660254037844386), RR(0.
˓→86602540378443871))))*QQbar(4*I)*(5*QQbar.polynomial_root(AA.common_polynomial(x^4
˓→+ x^3 + x^2 + x + 1), CIF(RIF(RR(0.3090169943749474), RR(0.30901699437494745)),
˓→RIF(RR(0.95105651629515353), RR(0.95105651629515364)))))
Note that the verify=True argument to sage_input will always trigger exact computation, so running
sage_input twice in a row on the same number will actually give different answers. In the following, running
sage_input on n will also trigger exact computation on rt2 , as you can see by the fact that the third output is
different than the first:
sage: rt2 = AA(sqrt(2))
sage: n = rt2^2
sage: sage_input(n, verify=True)
# Verified
R.<x> = AA[]
v = AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.4142135623730949),
˓→RR(1.4142135623730951)))
v*v
sage: sage_input(n, verify=True)
# Verified
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AA(2)
sage: n = rt2^2
sage: sage_input(n, verify=True)
# Verified
AA(2)
Just for fun, let’s try sage_input on a very complicated expression. The output of this example changed with the
rewriting of polynomial multiplication algorithms in trac ticket #10255:
sage: rt2 = sqrt(AA(2))
sage: rt3 = sqrt(QQbar(3))
sage: x = polygen(QQbar)
sage: nrt3 = AA.polynomial_root((x-rt2)*(x+rt3), RIF(-2, -1))
sage: one = AA.polynomial_root((x-rt2)*(x-rt3)*(x-nrt3)*(x-1-rt3-nrt3), RIF(0.9, 1.1))
sage: one
1.000000000000000?
sage: sage_input(one, verify=True)
# Verified
R1 = QQbar['x']
x1 = R1.gen()
R2 = AA['x']
x2 = R2.gen()
cp1 = AA.common_polynomial(x2^2 - 2)
v1 = QQbar.polynomial_root(cp1, RIF(RR(1.4142135623730949), RR(1.4142135623730951)))
v2 = QQbar.polynomial_root(AA.common_polynomial(x1^2 - 3), CIF(RIF(RR(1.
˓→7320508075688772), RR(1.7320508075688774)), RIF(RR(0))))
v3 = -v1 - v2
v4 = QQbar.polynomial_root(cp1, RIF(RR(1.4142135623730949), RR(1.4142135623730951)))
cp2 = AA.common_polynomial(x1^2 + (-v4 + v2)*x1 - v4*v2)
v5 = QQbar.polynomial_root(cp2, RIF(-RR(1.7320508075688774), -RR(1.7320508075688772)))
v6 = v3 - v5
v7 = -1 - v2 - QQbar.polynomial_root(cp2, RIF(-RR(1.7320508075688774), -RR(1.
˓→7320508075688772)))
v8 = v1*v2
v9 = v8 - v3*v5
si = v8*v5
AA.polynomial_root(AA.common_polynomial(x1^4 + (v6 + v7)*x1^3 + (v9 + v6*v7)*x1^2 + (˓→si + v9*v7)*x1 - si*v7), RIF(RR(0.99999999999999989), RR(1.0000000000000002)))
sage: one
1
We can pickle and unpickle algebraic fields (and they are globally unique):
sage: loads(dumps(AlgebraicField())) is AlgebraicField()
True
sage: loads(dumps(AlgebraicRealField())) is AlgebraicRealField()
True
We can pickle and unpickle algebraic numbers:
sage: loads(dumps(QQbar(10))) == QQbar(10)
True
sage: loads(dumps(QQbar(5/2))) == QQbar(5/2)
True
sage: loads(dumps(QQbar.zeta(5))) == QQbar.zeta(5)
True
sage: t = QQbar(sqrt(2)); type(t._descr)
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<class 'sage.rings.qqbar.ANRoot'>
sage: loads(dumps(t)) == QQbar(sqrt(2))
True
sage: t.exactify(); type(t._descr)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: loads(dumps(t)) == QQbar(sqrt(2))
True
sage: t = ~QQbar(sqrt(2)); type(t._descr)
<class 'sage.rings.qqbar.ANUnaryExpr'>
sage: loads(dumps(t)) == 1/QQbar(sqrt(2))
True
sage: t = QQbar(sqrt(2)) + QQbar(sqrt(3)); type(t._descr)
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: loads(dumps(t)) == QQbar(sqrt(2)) + QQbar(sqrt(3))
True
We can convert elements of QQbar and AA into the following types: float , complex , RDF , CDF , RR , CC ,
RIF , CIF , ZZ , and QQ , with a few exceptions. (For the arbitrary-precision types, RR , CC , RIF , and CIF , it can
convert into a field of arbitrary precision.)
Converting from QQbar to a real type (float , RDF , RR , RIF , ZZ , or QQ ) succeeds only if the QQbar is actually
real (has an imaginary component of exactly zero). Converting from either AA or QQbar to ZZ or QQ succeeds only
if the number actually is an integer or rational. If conversion fails, a ValueError will be raised.
Here are examples of all of these conversions:
sage: all_vals = [AA(42), AA(22/7), AA(golden_ratio), QQbar(-13), QQbar(89/55),
˓→QQbar(-sqrt(7)), QQbar.zeta(5)]
sage: def convert_test_all(ty):
....:
def convert_test(v):
....:
try:
....:
return ty(v)
....:
except (TypeError, ValueError):
....:
return None
....:
return [convert_test(_) for _ in all_vals]
sage: convert_test_all(float)
[42.0, 3.1428571428571432, 1.618033988749895, -13.0, 1.6181818181818182, -2.
˓→6457513110645907, None]
sage: convert_test_all(complex)
[(42+0j), (3.1428571428571432+0j), (1.618033988749895+0j), (-13+0j), (1.
˓→6181818181818182+0j), (-2.6457513110645907+0j), (0.30901699437494745+0.
˓→9510565162951536j)]
sage: convert_test_all(RDF)
[42.0, 3.1428571428571432, 1.618033988749895, -13.0, 1.6181818181818182, -2.
˓→6457513110645907, None]
sage: convert_test_all(CDF)
[42.0, 3.1428571428571432, 1.618033988749895, -13.0, 1.6181818181818182, -2.
˓→6457513110645907, 0.30901699437494745 + 0.9510565162951536*I]
sage: convert_test_all(RR)
[42.0000000000000, 3.14285714285714, 1.61803398874989, -13.0000000000000, 1.
˓→61818181818182, -2.64575131106459, None]
sage: convert_test_all(CC)
[42.0000000000000, 3.14285714285714, 1.61803398874989, -13.0000000000000, 1.
˓→61818181818182, -2.64575131106459, 0.309016994374947 + 0.951056516295154*I]
sage: convert_test_all(RIF)
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[42, 3.142857142857143?, 1.618033988749895?, -13, 1.618181818181819?, -2.
˓→645751311064591?, None]
sage: convert_test_all(CIF)
[42, 3.142857142857143?, 1.618033988749895?, -13, 1.618181818181819?, -2.
˓→645751311064591?, 0.3090169943749474? + 0.9510565162951536?*I]
sage: convert_test_all(ZZ)
[42, None, None, -13, None, None, None]
sage: convert_test_all(QQ)
[42, 22/7, None, -13, 89/55, None, None]
Compute the exact coordinates of a 34-gon (the formulas used are from Weisstein, Eric W. “Trigonometry Angles–
Pi/17.” and can be found at http://mathworld.wolfram.com/TrigonometryAnglesPi17.html):
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
sage:
rt17 = AA(17).sqrt()
rt2 = AA(2).sqrt()
eps = (17 + rt17).sqrt()
epss = (17 - rt17).sqrt()
delta = rt17 - 1
alpha = (34 + 6*rt17 + rt2*delta*epss - 8*rt2*eps).sqrt()
beta = 2*(17 + 3*rt17 - 2*rt2*eps - rt2*epss).sqrt()
x = rt2*(15 + rt17 + rt2*(alpha + epss)).sqrt()/8
y = rt2*(epss**2 - rt2*(alpha + epss)).sqrt()/8
sage: cx, cy = 1, 0
sage: for i in range(34):
....:
cx, cy = x*cx-y*cy, x*cy+y*cx
sage: cx
1.000000000000000?
sage: cy
0.?e-15
sage: ax = polygen(AA)
sage: x2 = AA.polynomial_root(256*ax**8 - 128*ax**7 - 448*ax**6 + 192*ax**5 +
˓→240*ax**4 - 80*ax**3 - 40*ax**2 + 8*ax + 1, RIF(0.9829, 0.983))
sage: y2 = (1-x2**2).sqrt()
sage: x - x2
0.?e-18
sage: y - y2
0.?e-17
Ideally, in the above example we should be able to test x == x2 and y == y2 but this is currently infinitely long.
class sage.rings.qqbar. ANBinaryExpr ( left, right, op)
Bases: sage.rings.qqbar.ANDescr
Initialize this ANBinaryExpr.
EXAMPLES:
sage: t = QQbar(sqrt(2)) + QQbar(sqrt(3)); type(t._descr) # indirect doctest
<class 'sage.rings.qqbar.ANBinaryExpr'>
exactify ( )
handle_sage_input ( sib, coerce, is_qqbar)
Produce an expression which will reproduce this value when evaluated, and an indication of whether this
value is worth sharing (always True for ANBinaryExpr ).
EXAMPLES:
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sage: sage_input(2 + sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
2 + AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(sqrt(AA(2)) + 2, verify=True)
# Verified
R.<x> = AA[]
AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.4142135623730949),
˓→RR(1.4142135623730951))) + 2
sage: sage_input(2 - sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
2 - AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(2 / sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
2/AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(2 + (-1*sqrt(AA(2))), verify=True)
# Verified
R.<x> = AA[]
2 - AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(2*sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
2*AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: rt2 = sqrt(AA(2))
sage: one = rt2/rt2
sage: n = one+3
sage: sage_input(n)
R.<x> = AA[]
v = AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
v/v + 3
sage: one == 1
True
sage: sage_input(n)
1 + AA(3)
sage: rt3 = QQbar(sqrt(3))
sage: one = rt3/rt3
sage: n = sqrt(AA(2))+one
sage: one == 1
True
sage: sage_input(n)
R.<x> = AA[]
QQbar.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951))) + 1
sage: from sage.rings.qqbar import *
sage: from sage.misc.sage_input import SageInputBuilder
sage: sib = SageInputBuilder()
sage: binexp = ANBinaryExpr(AA(3), AA(5), operator.mul)
sage: binexp.handle_sage_input(sib, False, False)
({binop:* {atomic:3} {call: {atomic:AA}({atomic:5})}}, True)
sage: binexp.handle_sage_input(sib, False, True)
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({call: {atomic:QQbar}({binop:* {atomic:3} {call: {atomic:AA}({atomic:5})}})}
˓→, True)
is_complex ( )
Whether this element is complex. Does not trigger exact computation, so may return True even if the
element is real.
EXAMPLES:
sage: x = (QQbar(sqrt(-2)) / QQbar(sqrt(-5)))._descr
sage: x.is_complex()
True
class sage.rings.qqbar. ANDescr
Bases: sage.structure.sage_object.SageObject
An AlgebraicNumber or AlgebraicReal is a wrapper around an ANDescr object. ANDescr is
an abstract base class, which should never be directly instantiated; its concrete subclasses are ANRational
, ANBinaryExpr , ANUnaryExpr , ANRoot , and ANExtensionElement . ANDescr and all of its
subclasses are for internal use, and should not be used directly.
abs ( n)
Absolute value of self.
EXAMPLES:
sage: a = QQbar(sqrt(2))
sage: b = a._descr
sage: b.abs(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
conjugate ( n)
Complex conjugate of self.
EXAMPLES:
sage: a = QQbar(sqrt(-7))
sage: b = a._descr
sage: b.conjugate(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
imag ( n)
Imaginary part of self.
EXAMPLES:
sage: a = QQbar(sqrt(-7))
sage: b = a._descr
sage: b.imag(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
invert ( n)
1/self.
EXAMPLES:
sage: a = QQbar(sqrt(2))
sage: b = a._descr
sage: b.invert(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
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is_simple ( )
Checks whether this descriptor represents a value with the same algebraic degree as the number field
associated with the descriptor.
Returns True if self is an ANRational , or a minimal ANExtensionElement .
EXAMPLES:
sage:
sage:
True
sage:
sage:
sage:
sage:
sage:
True
sage:
sage:
False
sage:
sage:
True
from sage.rings.qqbar import ANRational
ANRational(1/2).is_simple()
rt2 = AA(sqrt(2))
rt3 = AA(sqrt(3))
rt2b = rt3 + rt2 - rt3
rt2.exactify()
rt2._descr.is_simple()
rt2b.exactify()
rt2b._descr.is_simple()
rt2b.simplify()
rt2b._descr.is_simple()
neg ( n)
Negation of self.
EXAMPLES:
sage: a = QQbar(sqrt(2))
sage: b = a._descr
sage: b.neg(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
norm ( n)
Field norm of self from Q to its real subfield A, i.e.~the square of the usual complex absolute value.
EXAMPLES:
sage: a = QQbar(sqrt(-7))
sage: b = a._descr
sage: b.norm(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
real ( n)
Real part of self.
EXAMPLES:
sage: a = QQbar(sqrt(-7))
sage: b = a._descr
sage: b.real(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
class sage.rings.qqbar. ANExtensionElement ( generator, value)
Bases: sage.rings.qqbar.ANDescr
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The subclass of ANDescr that represents a number field element in terms of a specific generator. Consists of a polynomial with rational coefficients in terms of the generator, and the generator itself, an
AlgebraicGenerator .
abs ( n)
Return the absolute value of self (square root of the norm).
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(-3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.abs(a)
Root 3.146264369941972342? of x^2 - 9.89897948556636?
conjugate ( n)
Negation of self.
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(-3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.conjugate(a)
-1/3*a^3 + 2/3*a^2 - 4/3*a + 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in
˓→-0.7247448713915890? + 1.573132184970987?*I
sage: b.conjugate("ham spam and eggs")
-1/3*a^3 + 2/3*a^2 - 4/3*a + 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in
˓→-0.7247448713915890? + 1.573132184970987?*I
exactify ( )
Return an exact representation of self. Since self is already exact, just return self.
EXAMPLES:
sage: v = (x^2 - x - 1).roots(ring=AA, multiplicities=False)[1]._descr.
˓→exactify()
sage: type(v)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: v.exactify() is v
True
field_element_value ( )
Return the underlying number field element.
EXAMPLES:
sage: v = (x^2 - x - 1).roots(ring=AA, multiplicities=False)[1]._descr.
˓→exactify()
sage: v.field_element_value()
a
generator ( )
Return the AlgebraicGenerator object corresponding to self.
EXAMPLES:
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sage: v = (x^2 - x - 1).roots(ring=AA, multiplicities=False)[1]._descr.
˓→exactify()
sage: v.generator()
Number Field in a with defining polynomial y^2 - y - 1 with a in 1.
˓→618033988749895?
handle_sage_input ( sib, coerce, is_qqbar)
Produce an expression which will reproduce this value when evaluated, and an indication of whether this
value is worth sharing (always True, for ANExtensionElement ).
EXAMPLES:
sage: I = QQbar(I)
sage: sage_input(3+4*I, verify=True)
# Verified
QQbar(3 + 4*I)
sage: v = QQbar.zeta(3) + QQbar.zeta(5)
sage: v - v == 0
True
sage: sage_input(vector(QQbar, (4-3*I, QQbar.zeta(7))), verify=True)
# Verified
R.<x> = AA[]
vector(QQbar, [4 - 3*I, QQbar.polynomial_root(AA.common_polynomial(x^6 + x^5
˓→+ x^4 + x^3 + x^2 + x + 1), CIF(RIF(RR(0.62348980185873348), RR(0.
˓→62348980185873359)), RIF(RR(0.7818314824680298), RR(0.
˓→78183148246802991))))])
sage: sage_input(v, verify=True)
# Verified
R.<x> = AA[]
v = QQbar.polynomial_root(AA.common_polynomial(x^8 - x^7 + x^5 - x^4 + x^3 ˓→x + 1), CIF(RIF(RR(0.91354545764260087), RR(0.91354545764260098)), RIF(RR(0.
˓→40673664307580015), RR(0.40673664307580021))))
v^5 + v^3
sage: v = QQbar(sqrt(AA(2)))
sage: v.exactify()
sage: sage_input(v, verify=True)
# Verified
R.<x> = AA[]
QQbar(AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951))))
sage: from sage.rings.qqbar import *
sage: from sage.misc.sage_input import SageInputBuilder
sage: sib = SageInputBuilder()
sage: extel = ANExtensionElement(QQbar_I_generator, QQbar_I_generator.field().
˓→gen() + 1)
sage: extel.handle_sage_input(sib, False, True)
({call: {atomic:QQbar}({binop:+ {atomic:1} {atomic:I}})}, True)
invert ( n)
1/self.
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(-3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
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sage: b.invert(a)
7/3*a^3 - 2/3*a^2 + 4/3*a - 12 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in
˓→-0.7247448713915890? - 1.573132184970987?*I
sage: b.invert("ham spam and eggs")
7/3*a^3 - 2/3*a^2 + 4/3*a - 12 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in
˓→-0.7247448713915890? - 1.573132184970987?*I
is_complex ( )
Return True if the number field that defines this element is not real. This does not imply that the element
itself is definitely non-real, as in the example below.
EXAMPLES:
sage: rt2 = QQbar(sqrt(2))
sage: rtm3 = QQbar(sqrt(-3))
sage: x = rtm3 + rt2 - rtm3
sage: x.exactify()
sage: y = x._descr
sage: type(y)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: y.is_complex()
True
sage: x.imag() == 0
True
is_simple ( )
Checks whether this descriptor represents a value with the same algebraic degree as the number field
associated with the descriptor.
For ANExtensionElement elements, we check this by comparing the degree of the minimal polynomial to the degree of the field.
EXAMPLES:
sage: rt2 = AA(sqrt(2))
sage: rt3 = AA(sqrt(3))
sage: rt2b = rt3 + rt2 - rt3
sage: rt2.exactify()
sage: rt2._descr
a where a^2 - 2 = 0 and a in 1.414213562373095?
sage: rt2._descr.is_simple()
True
sage:
sage:
a^3 sage:
False
rt2b.exactify()
rt2b._descr
3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
rt2b._descr.is_simple()
minpoly ( )
Compute the minimal polynomial of this algebraic number.
EXAMPLES:
sage: v = (x^2 - x - 1).roots(ring=AA, multiplicities=False)[1]._descr.
˓→exactify()
sage: type(v)
<class 'sage.rings.qqbar.ANExtensionElement'>
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sage: v.minpoly()
x^2 - x - 1
neg ( n)
Negation of self.
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(-3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.neg(a)
1/3*a^3 - 2/3*a^2 + 4/3*a - 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in ˓→0.7247448713915890? - 1.573132184970987?*I
sage: b.neg("ham spam and eggs")
1/3*a^3 - 2/3*a^2 + 4/3*a - 2 where a^4 - 2*a^3 + a^2 - 6*a + 9 = 0 and a in ˓→0.7247448713915890? - 1.573132184970987?*I
norm ( n)
Norm of self (square of complex absolute value)
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(-3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.norm(a)
<class 'sage.rings.qqbar.ANUnaryExpr'>
rational_argument ( n)
If the argument of self is 2𝜋 times some rational number in [1/2, −1/2), return that rational; otherwise,
return None .
EXAMPLES:
sage: a = QQbar(sqrt(-2)) + QQbar(sqrt(3))
sage: a.exactify()
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: b.rational_argument(a) is None
True
sage: x = polygen(QQ)
sage: a = (x^4 + 1).roots(QQbar, multiplicities=False)[0]
sage: a.exactify()
sage: b = a._descr
sage: b.rational_argument(a)
-3/8
simplify ( n)
Compute an exact representation for this descriptor, in the smallest possible number field.
INPUT:
•n – The element of AA or QQbar corresponding to this descriptor.
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EXAMPLES:
sage: rt2 = AA(sqrt(2))
sage: rt3 = AA(sqrt(3))
sage: rt2b = rt3 + rt2 - rt3
sage: rt2b.exactify()
sage: rt2b._descr
a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
sage: rt2b._descr.simplify(rt2b)
a where a^2 - 2 = 0 and a in 1.414213562373095?
class sage.rings.qqbar. ANRational ( x)
Bases: sage.rings.qqbar.ANDescr
The subclass of ANDescr that represents an arbitrary rational. This class is private, and should not be used
directly.
abs ( n)
Absolute value of self.
EXAMPLES:
sage: a = QQbar(3)
sage: b = a._descr
sage: b.abs(a)
3
angle ( )
Return a rational number 𝑞 ∈ (−1/2, 1/2] such that self is a rational multiple of 𝑒2𝜋𝑖𝑞 . Always returns
0, since this element is rational.
EXAMPLES:
sage: QQbar(3)._descr.angle()
0
sage: QQbar(-3)._descr.angle()
0
sage: QQbar(0)._descr.angle()
0
exactify ( )
Calculate self exactly. Since self is a rational number, return self.
EXAMPLES:
sage: a = QQbar(1/3)._descr
sage: a.exactify() is a
True
generator ( )
Return an AlgebraicGenerator object associated to this element. Returns the trivial generator, since
self is rational.
EXAMPLES:
sage: QQbar(0)._descr.generator()
Trivial generator
handle_sage_input ( sib, coerce, is_qqbar)
Produce an expression which will reproduce this value when evaluated, and an indication of whether this
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value is worth sharing (always False, for rationals).
EXAMPLES:
sage: sage_input(QQbar(22/7), verify=True)
# Verified
QQbar(22/7)
sage: sage_input(-AA(3)/5, verify=True)
# Verified
AA(-3/5)
sage: sage_input(vector(AA, (0, 1/2, 1/3)), verify=True)
# Verified
vector(AA, [0, 1/2, 1/3])
sage: from sage.rings.qqbar import *
sage: from sage.misc.sage_input import SageInputBuilder
sage: sib = SageInputBuilder()
sage: rat = ANRational(9/10)
sage: rat.handle_sage_input(sib, False, True)
({call: {atomic:QQbar}({binop:/ {atomic:9} {atomic:10}})}, False)
invert ( n)
1/self.
EXAMPLES:
sage: a = QQbar(3)
sage: b = a._descr
sage: b.invert(a)
1/3
is_complex ( )
Return False, since rational numbers are real
EXAMPLES:
sage: QQbar(1/7)._descr.is_complex()
False
is_simple ( )
Checks whether this descriptor represents a value with the same algebraic degree as the number field
associated with the descriptor.
This is always true for rational numbers.
EXAMPLES:
sage: AA(1/2)._descr.is_simple()
True
minpoly ( )
Return the min poly of self over Q.
EXAMPLES:
sage: QQbar(7)._descr.minpoly()
x - 7
neg ( n)
Negation of self.
EXAMPLES:
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sage: a = QQbar(3)
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANRational'>
sage: b.neg(a)
-3
rational_argument ( n)
Return the argument of self divided by 2𝜋, or None if this element is 0.
EXAMPLES:
sage: QQbar(3)._descr.rational_argument(None)
0
sage: QQbar(-3)._descr.rational_argument(None)
1/2
sage: QQbar(0)._descr.rational_argument(None) is None
True
scale ( )
Return a rational number 𝑟 such that self is equal to 𝑟𝑒2𝜋𝑖𝑞 for some 𝑞 ∈ (−1/2, 1/2]. In other words,
just return self as a rational number.
EXAMPLES:
sage: QQbar(-3)._descr.scale()
-3
class sage.rings.qqbar. ANRoot ( poly, interval, multiplicity=1)
Bases: sage.rings.qqbar.ANDescr
The subclass of ANDescr that represents a particular root of a polynomial with algebraic coefficients. This
class is private, and should not be used directly.
conjugate ( n)
Complex conjugate of this ANRoot object.
EXAMPLES:
sage: a = (x^2 + 23).roots(ring=QQbar, multiplicities=False)[0]
sage: b = a._descr
sage: type(b)
<class 'sage.rings.qqbar.ANRoot'>
sage: c = b.conjugate(a); c
<class 'sage.rings.qqbar.ANUnaryExpr'>
sage: c.exactify()
-2*a + 1 where a^2 - a + 6 = 0 and a in 0.50000000000000000? - 2.
˓→397915761656360?*I
exactify ( )
Returns either an ANRational or an ANExtensionElement with the same value as this number.
EXAMPLES:
sage:
sage:
sage:
sage:
2
from sage.rings.qqbar import ANRoot
x = polygen(QQbar)
two = ANRoot((x-2)*(x-sqrt(QQbar(2))), RIF(1.9, 2.1))
two.exactify()
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sage: strange = ANRoot(x^2 + sqrt(QQbar(3))*x - sqrt(QQbar(2)), RIF(-0, 1))
sage: strange.exactify()
a where a^8 - 6*a^6 + 5*a^4 - 12*a^2 + 4 = 0 and a in 0.6051012265139511?
handle_sage_input ( sib, coerce, is_qqbar)
Produce an expression which will reproduce this value when evaluated, and an indication of whether this
value is worth sharing (always True, for ANRoot ).
EXAMPLES:
sage: sage_input((AA(3)^(1/2))^(1/3), verify=True)
# Verified
R.<x> = AA[]
AA.polynomial_root(AA.common_polynomial(x^3 - AA.polynomial_root(AA.common_
˓→polynomial(x^2 - 3), RIF(RR(1.7320508075688772), RR(1.7320508075688774)))),
˓→RIF(RR(1.2009369551760025), RR(1.2009369551760027)))
These two examples are too big to verify quickly. (Verification would create a field of degree 28.):
sage: sage_input((sqrt(AA(3))^(5/7))^(9/4))
R.<x> = AA[]
v1 = AA.polynomial_root(AA.common_polynomial(x^2 - 3), RIF(RR(1.
˓→7320508075688772), RR(1.7320508075688774)))
v2 = v1*v1
v3 = AA.polynomial_root(AA.common_polynomial(x^7 - v2*v2*v1), RIF(RR(1.
˓→4804728524798112), RR(1.4804728524798114)))
v4 = v3*v3
v5 = v4*v4
AA.polynomial_root(AA.common_polynomial(x^4 - v5*v5*v3), RIF(RR(2.
˓→4176921938267877), RR(2.4176921938267881)))
sage: sage_input((sqrt(QQbar(-7))^(5/7))^(9/4))
R.<x> = QQbar[]
v1 = QQbar.polynomial_root(AA.common_polynomial(x^2 + 7), CIF(RIF(RR(0)),
˓→RIF(RR(2.6457513110645903), RR(2.6457513110645907))))
v2 = v1*v1
v3 = QQbar.polynomial_root(AA.common_polynomial(x^7 - v2*v2*v1), CIF(RIF(RR(0.
˓→8693488875796217), RR(0.86934888757962181)), RIF(RR(1.8052215661454434),
˓→RR(1.8052215661454436))))
v4 = v3*v3
v5 = v4*v4
QQbar.polynomial_root(AA.common_polynomial(x^4 - v5*v5*v3), CIF(RIF(-RR(3.
˓→8954086044650791), -RR(3.8954086044650786)), RIF(RR(2.7639398015408925),
˓→RR(2.7639398015408929))))
sage: x = polygen(QQ)
sage: sage_input(AA.polynomial_root(x^2-x-1, RIF(1, 2)), verify=True)
# Verified
R.<x> = AA[]
AA.polynomial_root(AA.common_polynomial(x^2 - x - 1), RIF(RR(1.
˓→6180339887498947), RR(1.6180339887498949)))
sage: sage_input(QQbar.polynomial_root(x^3-5, CIF(RIF(-3, 0), RIF(0, 3))),
˓→verify=True)
# Verified
R.<x> = AA[]
QQbar.polynomial_root(AA.common_polynomial(x^3 - 5), CIF(RIF(-RR(0.
˓→85498797333834853), -RR(0.85498797333834842)), RIF(RR(1.4808826096823642),
˓→RR(1.4808826096823644))))
sage: from sage.rings.qqbar import *
sage: from sage.misc.sage_input import SageInputBuilder
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sage: sib = SageInputBuilder()
sage: rt = ANRoot(x^3 - 2, RIF(0, 4))
sage: rt.handle_sage_input(sib, False, True)
({call: {getattr: {atomic:QQbar}.polynomial_root}({call: {getattr: {atomic:AA}
˓→.common_polynomial}({binop:- {binop:** {gen:x {constr_parent: {subscr:
˓→{atomic:AA}[{atomic:'x'}]} with gens: ('x',)}} {atomic:3}} {atomic:2}})},
˓→{call: {atomic:RIF}({call: {atomic:RR}({atomic:1.259921049894873})}, {call:
˓→{atomic:RR}({atomic:1.2599210498948732})})})}, True)
is_complex ( )
Whether this is a root in Q (rather than A). Note that this may return True even if the root is actually real,
as the second example shows; it does not trigger exact computation to see if the root is real.
EXAMPLES:
sage: x = polygen(QQ)
sage: (x^2 - x - 1).roots(ring=AA, multiplicities=False)[1]._descr.is_
˓→complex()
False
sage: (x^2 - x - 1).roots(ring=QQbar, multiplicities=False)[1]._descr.is_
˓→complex()
True
refine_interval ( interval, prec)
Takes an interval which is assumed to enclose exactly one root of the polynomial (or, with multiplicity=‘k‘,
exactly one root of the 𝑘 − 1-st derivative); and a precision, in bits.
Tries to find a narrow interval enclosing the root using interval arithmetic of the given precision. (No
particular number of resulting bits of precision is guaranteed.)
Uses a combination of Newton’s method (adapted for interval arithmetic) and bisection. The algorithm
will converge very quickly if started with a sufficiently narrow interval.
EXAMPLES:
sage: from sage.rings.qqbar import ANRoot
sage: x = polygen(AA)
sage: rt2 = ANRoot(x^2 - 2, RIF(0, 2))
sage: rt2.refine_interval(RIF(0, 2), 75)
1.4142135623730950488017?
class sage.rings.qqbar. ANRootOfUnity ( generator, value)
Bases: sage.rings.qqbar.ANExtensionElement
Deprecated class to support unpickling
class sage.rings.qqbar. ANUnaryExpr ( arg, op)
Bases: sage.rings.qqbar.ANDescr
Initialize this ANUnaryExpr.
EXAMPLES:
sage: t = ~QQbar(sqrt(2)); type(t._descr) # indirect doctest
<class 'sage.rings.qqbar.ANUnaryExpr'>
exactify ( )
Trigger exact computation of self.
EXAMPLES:
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sage: v = (-QQbar(sqrt(2)))._descr
sage: type(v)
<class 'sage.rings.qqbar.ANUnaryExpr'>
sage: v.exactify()
-a where a^2 - 2 = 0 and a in 1.414213562373095?
handle_sage_input ( sib, coerce, is_qqbar)
Produce an expression which will reproduce this value when evaluated, and an indication of whether this
value is worth sharing (always True for ANUnaryExpr ).
EXAMPLES:
sage: sage_input(-sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
-AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(~sqrt(AA(2)), verify=True)
# Verified
R.<x> = AA[]
~AA.polynomial_root(AA.common_polynomial(x^2 - 2), RIF(RR(1.
˓→4142135623730949), RR(1.4142135623730951)))
sage: sage_input(sqrt(QQbar(-3)).conjugate(), verify=True)
# Verified
R.<x> = QQbar[]
QQbar.polynomial_root(AA.common_polynomial(x^2 + 3), CIF(RIF(RR(0)), RIF(RR(1.
˓→7320508075688772), RR(1.7320508075688774)))).conjugate()
sage: sage_input(QQbar.zeta(3).real(), verify=True)
# Verified
R.<x> = AA[]
QQbar.polynomial_root(AA.common_polynomial(x^2 + x + 1), CIF(RIF(-RR(0.
˓→50000000000000011), -RR(0.49999999999999994)), RIF(RR(0.8660254037844386),
˓→RR(0.86602540378443871)))).real()
sage: sage_input(QQbar.zeta(3).imag(), verify=True)
# Verified
R.<x> = AA[]
QQbar.polynomial_root(AA.common_polynomial(x^2 + x + 1), CIF(RIF(-RR(0.
˓→50000000000000011), -RR(0.49999999999999994)), RIF(RR(0.8660254037844386),
˓→RR(0.86602540378443871)))).imag()
sage: sage_input(abs(sqrt(QQbar(-3))), verify=True)
# Verified
R.<x> = QQbar[]
abs(QQbar.polynomial_root(AA.common_polynomial(x^2 + 3), CIF(RIF(RR(0)),
˓→RIF(RR(1.7320508075688772), RR(1.7320508075688774)))))
sage: sage_input(sqrt(QQbar(-3)).norm(), verify=True)
# Verified
R.<x> = QQbar[]
QQbar.polynomial_root(AA.common_polynomial(x^2 + 3), CIF(RIF(RR(0)), RIF(RR(1.
˓→7320508075688772), RR(1.7320508075688774)))).norm()
sage: sage_input(QQbar(QQbar.zeta(3).real()), verify=True)
# Verified
R.<x> = AA[]
QQbar(QQbar.polynomial_root(AA.common_polynomial(x^2 + x + 1), CIF(RIF(-RR(0.
˓→50000000000000011), -RR(0.49999999999999994)), RIF(RR(0.8660254037844386),
˓→RR(0.86602540378443871)))).real())
sage: from sage.rings.qqbar import *
sage: from sage.misc.sage_input import SageInputBuilder
sage: sib = SageInputBuilder()
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sage: unexp = ANUnaryExpr(sqrt(AA(2)), '~')
sage: unexp.handle_sage_input(sib, False, False)
({unop:~ {call: {getattr: {atomic:AA}.polynomial_root}({call: {getattr:
˓→{atomic:AA}.common_polynomial}({binop:- {binop:** {gen:x {constr_parent:
˓→{subscr: {atomic:AA}[{atomic:'x'}]} with gens: ('x',)}} {atomic:2}} {atomic:
˓→2}})}, {call: {atomic:RIF}({call: {atomic:RR}({atomic:1.4142135623730949})}
˓→, {call: {atomic:RR}({atomic:1.4142135623730951})})})}},
True)
sage: unexp.handle_sage_input(sib, False, True)
({call: {atomic:QQbar}({unop:~ {call: {getattr: {atomic:AA}.polynomial_root}(
˓→{call: {getattr: {atomic:AA}.common_polynomial}({binop:- {binop:** {gen:x
˓→{constr_parent: {subscr: {atomic:AA}[{atomic:'x'}]} with gens: ('x',)}}
˓→{atomic:2}} {atomic:2}})}, {call: {atomic:RIF}({call: {atomic:RR}({atomic:1.
˓→4142135623730949})}, {call: {atomic:RR}({atomic:1.4142135623730951})})})}})}
˓→,
True)
is_complex ( )
Return whether or not this element is complex. Note that this is a data type check, and triggers no computations – if it returns False, the element might still be real, it just doesn’t know it yet.
EXAMPLES:
sage: t = AA(sqrt(2))
sage: s = (-t)._descr
sage: s
<class 'sage.rings.qqbar.ANUnaryExpr'>
sage: s.is_complex()
False
sage: QQbar(-sqrt(2))._descr.is_complex()
True
class sage.rings.qqbar. AlgebraicField
Bases: sage.misc.fast_methods.Singleton , sage.rings.qqbar.AlgebraicField_common
The field of all algebraic complex numbers.
algebraic_closure ( )
Return the algebraic closure of this field. As this field is already algebraically closed, just returns self.
EXAMPLES:
sage: QQbar.algebraic_closure()
Algebraic Field
completion ( p, prec, extras={})
Return the completion of self at the place 𝑝. Only implemented for 𝑝 = ∞ at present.
INPUT:
•p – either a prime (not implemented at present) or Infinity
•prec – precision of approximate field to return
•extras – a dict of extra keyword arguments for the RealField constructor
EXAMPLES:
sage: QQbar.completion(infinity, 500)
Complex Field with 500 bits of precision
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sage: QQbar.completion(infinity, prec=53, extras={'type':'RDF'})
Complex Double Field
sage: QQbar.completion(infinity, 53) is CC
True
sage: QQbar.completion(3, 20)
Traceback (most recent call last):
...
NotImplementedError
construction ( )
Return a functor that constructs self (used by the coercion machinery).
EXAMPLES:
sage: QQbar.construction()
(AlgebraicClosureFunctor, Rational Field)
gen ( n=0)
Return the 𝑛-th element of the tuple returned by gens() .
EXAMPLES:
sage: QQbar.gen(0)
I
sage: QQbar.gen(1)
Traceback (most recent call last):
...
IndexError: n must be 0
gens ( )
Return a set of generators for this field. As this field is not finitely generated over its prime field, we opt
for just returning I.
EXAMPLES:
sage: QQbar.gens()
(I,)
ngens ( )
Return the size of the tuple returned by gens() .
EXAMPLES:
sage: QQbar.ngens()
1
polynomial_root ( poly, interval, multiplicity=1)
Given a polynomial with algebraic coefficients and an interval enclosing exactly one root of the polynomial, constructs an algebraic real representation of that root.
The polynomial need not be irreducible, or even squarefree; but if the given root is a multiple root, its
multiplicity must be specified. (IMPORTANT NOTE: Currently, multiplicity-𝑘 roots are handled by taking
the (𝑘 − 1)-st derivative of the polynomial. This means that the interval must enclose exactly one root of
this derivative.)
The conditions on the arguments (that the interval encloses exactly one root, and that multiple roots match
the given multiplicity) are not checked; if they are not satisfied, an error may be thrown (possibly later,
when the algebraic number is used), or wrong answers may result.
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Note that if you are constructing multiple roots of a single polynomial, it is better to use
QQbar.common_polynomial to get a shared polynomial.
EXAMPLES:
sage: x = polygen(QQbar)
sage: phi = QQbar.polynomial_root(x^2 - x - 1, RIF(0, 2)); phi
1.618033988749895?
sage: p = (x-1)^7 * (x-2)
sage: r = QQbar.polynomial_root(p, RIF(9/10, 11/10), multiplicity=7)
sage: r; r == 1
1
True
sage: p = (x-phi)*(x-sqrt(QQbar(2)))
sage: r = QQbar.polynomial_root(p, RIF(1, 3/2))
sage: r; r == sqrt(QQbar(2))
1.414213562373095?
True
random_element ( poly_degree=2, *args, **kwds)
Returns a random algebraic number.
INPUT:
•poly_degree - default: 2 - degree of the random polynomial over the integers of which the returned
algebraic number is a root. This is not necessarily the degree of the minimal polynomial of the
number. Increase this parameter to achieve a greater diversity of algebraic numbers, at a cost of
greater computation time. You can also vary the distribution of the coefficients but that will not vary
the degree of the extension containing the element.
•args , kwds - arguments and keywords passed to the random number generator for elements of ZZ
, the integers. See random_element() for details, or see example below.
OUTPUT:
An element of QQbar , the field of algebraic numbers (see sage.rings.qqbar ).
ALGORITHM:
A polynomial with degree between 1 and poly_degree , with random integer coefficients is created. A
root of this polynomial is chosen at random. The default degree is 2 and the integer coefficients come from
a distribution heavily weighted towards 0, ±1, ±2.
EXAMPLES:
sage: a = QQbar.random_element()
sage: a
# random
0.2626138748742799? + 0.8769062830975992?*I
sage: a in QQbar
True
sage: b = QQbar.random_element(poly_degree=20)
sage: b
# random
-0.8642649077479498? - 0.5995098147478391?*I
sage: b in QQbar
True
Parameters for the distribution of the integer coefficients of the polynomials can be passed on to the random
element method for integers. For example, current default behavior of this method returns zero about 15%
of the time; if we do not include zero as a possible coefficient, there will never be a zero constant term,
and thus never a zero root.
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sage: z = [QQbar.random_element(x=1, y=10) for _ in range(20)]
sage: QQbar(0) in z
False
If you just want real algebraic numbers you can filter them out.
Using an odd degree for the polynomials will insure some degree of
success. ::
sage: r = []
sage: while len(r) < 3:
....:
x = QQbar.random_element(poly_degree=3)
....:
if x in AA:
....:
r.append(x)
sage: (len(r) == 3) and all([z in AA for z in r])
True
zeta ( n=4)
Returns a primitive 𝑛‘th root of unity, specifically exp(2 * 𝜋 * 𝑖/𝑛).
INPUT:
•n (integer) – default 4
EXAMPLES:
sage: QQbar.zeta(1)
1
sage: QQbar.zeta(2)
-1
sage: QQbar.zeta(3)
-0.500000000000000? + 0.866025403784439?*I
sage: QQbar.zeta(4)
I
sage: QQbar.zeta()
I
sage: QQbar.zeta(5)
0.3090169943749474? + 0.9510565162951536?*I
sage: QQbar.zeta(3000)
0.999997806755380? + 0.002094393571219374?*I
class sage.rings.qqbar. AlgebraicField_common
Bases: sage.rings.ring.Field
Common base class for the classes AlgebraicRealField and AlgebraicField .
characteristic ( )
Return the characteristic of this field. Since this class is only used for fields of characteristic 0, always
returns 0.
EXAMPLES:
sage: AA.characteristic()
0
common_polynomial ( poly)
Given a polynomial with algebraic coefficients, returns a wrapper that caches high-precision calculations
and factorizations. This wrapper can be passed to polynomial_root in place of the polynomial.
Using common_polynomial makes no semantic difference, but will improve efficiency if you are
dealing with multiple roots of a single polynomial.
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EXAMPLES:
sage:
sage:
sage:
sage:
sage:
True
sage:
True
x =
p =
phi
tau
phi
polygen(ZZ)
AA.common_polynomial(x^2 - x - 1)
= AA.polynomial_root(p, RIF(1, 2))
= AA.polynomial_root(p, RIF(-1, 0))
+ tau == 1
phi * tau == -1
sage: x = polygen(SR)
sage: p = (x - sqrt(-5)) * (x - sqrt(3)); p
x^2 + (-sqrt(3) - sqrt(-5))*x + sqrt(3)*sqrt(-5)
sage: p = QQbar.common_polynomial(p)
sage: a = QQbar.polynomial_root(p, CIF(RIF(-0.1, 0.1), RIF(2, 3))); a
0.?e-18 + 2.236067977499790?*I
sage: b = QQbar.polynomial_root(p, RIF(1, 2)); b
1.732050807568878?
These “common polynomials” can be shared between real and complex roots:
sage: p = AA.common_polynomial(x^3 - x - 1)
sage: r1 = AA.polynomial_root(p, RIF(1.3, 1.4)); r1
1.324717957244746?
sage: r2 = QQbar.polynomial_root(p, CIF(RIF(-0.7, -0.6), RIF(0.5, 0.6))); r2
-0.6623589786223730? + 0.5622795120623013?*I
default_interval_prec ( )
Return the default interval precision used for root isolation.
EXAMPLES:
sage: AA.default_interval_prec()
64
is_finite ( )
Check whether this field is finite. Since this class is only used for fields of characteristic 0, always returns
False.
EXAMPLES:
sage: QQbar.is_finite()
False
order ( )
Return the cardinality of self. Since this class is only used for fields of characteristic 0, always returns
Infinity.
EXAMPLES:
sage: QQbar.order()
+Infinity
class sage.rings.qqbar. AlgebraicGenerator ( field, root)
Bases: sage.structure.sage_object.SageObject
An AlgebraicGenerator represents both an algebraic number 𝛼 and the number field Q[𝛼]. There is a
single AlgebraicGenerator representing Q (with 𝛼 = 0).
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The AlgebraicGenerator class is private, and should not be used directly.
conjugate ( )
If this generator is for the algebraic number 𝛼, return a generator for the complex conjugate of 𝛼.
EXAMPLES:
sage: from sage.rings.qqbar import AlgebraicGenerator
sage: x = polygen(QQ); f = x^4 + x + 17
sage: nf = NumberField(f,name='a')
sage: b = f.roots(QQbar)[0][0]
sage: root = b._descr
sage: gen = AlgebraicGenerator(nf, root)
sage: gen.conjugate()
Number Field in a with defining polynomial x^4 + x + 17 with a in -1.
˓→436449997483091? + 1.374535713065812?*I
field ( )
Return the number field attached to self.
EXAMPLES:
sage: from sage.rings.qqbar import qq_generator
sage: qq_generator.field()
Rational Field
is_complex ( )
Return True if this is a generator for a non-real number field.
EXAMPLES:
sage: z7 = QQbar.zeta(7)
sage: g = z7._descr._generator
sage: g.is_complex()
True
sage:
sage:
sage:
sage:
sage:
sage:
sage:
False
from sage.rings.qqbar import ANRoot, AlgebraicGenerator
y = polygen(QQ, 'y')
x = polygen(QQbar)
nf = NumberField(y^2 - y - 1, name='a', check=False)
root = ANRoot(x^2 - x - 1, RIF(1, 2))
gen = AlgebraicGenerator(nf, root)
gen.is_complex()
is_trivial ( )
Returns true iff this is the trivial generator (alpha == 1), which does not actually extend the rationals.
EXAMPLES:
sage: from sage.rings.qqbar import qq_generator
sage: qq_generator.is_trivial()
True
pari_field ( )
Return the PARI field attached to this generator.
EXAMPLES:
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sage: from sage.rings.qqbar import qq_generator
sage: qq_generator.pari_field()
Traceback (most recent call last):
...
ValueError: No PARI field attached to trivial generator
sage:
sage:
sage:
sage:
sage:
sage:
sage:
[y^2
from sage.rings.qqbar import ANRoot, AlgebraicGenerator, qq_generator
y = polygen(QQ)
x = polygen(QQbar)
nf = NumberField(y^2 - y - 1, name='a', check=False)
root = ANRoot(x^2 - x - 1, RIF(1, 2))
gen = AlgebraicGenerator(nf, root)
gen.pari_field()
- y - 1, [2, 0], ...]
root_as_algebraic ( )
Return the root attached to self as an algebraic number.
EXAMPLES:
sage: t = sage.rings.qqbar.qq_generator.root_as_algebraic(); t
1
sage: t.parent()
Algebraic Real Field
super_poly ( super, checked=None)
Given a generator gen and another generator super , where super is the result of a tree of union()
operations where one of the leaves is gen , gen.super_poly(super) returns a polynomial expressing the value of gen in terms of the value of super (except that if gen is qq_generator ,
super_poly() always returns None.)
EXAMPLES:
sage: from sage.rings.qqbar import AlgebraicGenerator, ANRoot, qq_generator
sage: _.<y> = QQ['y']
sage: x = polygen(QQbar)
sage: nf2 = NumberField(y^2 - 2, name='a', check=False)
sage: root2 = ANRoot(x^2 - 2, RIF(1, 2))
sage: gen2 = AlgebraicGenerator(nf2, root2)
sage: gen2
Number Field in a with defining polynomial y^2 - 2 with a in 1.
˓→414213562373095?
sage: nf3 = NumberField(y^2 - 3, name='a', check=False)
sage: root3 = ANRoot(x^2 - 3, RIF(1, 2))
sage: gen3 = AlgebraicGenerator(nf3, root3)
sage: gen3
Number Field in a with defining polynomial y^2 - 3 with a in 1.
˓→732050807568878?
sage: gen2_3 = gen2.union(gen3)
sage: gen2_3
Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 0.
˓→5176380902050415?
sage: qq_generator.super_poly(gen2) is None
True
sage: gen2.super_poly(gen2_3)
-a^3 + 3*a
sage: gen3.super_poly(gen2_3)
-a^2 + 2
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union ( other)
Given generators alpha and beta , alpha.union(beta) gives a generator for the number field
Q[𝛼][𝛽].
EXAMPLES:
sage: from sage.rings.qqbar import ANRoot, AlgebraicGenerator, qq_generator
sage: _.<y> = QQ['y']
sage: x = polygen(QQbar)
sage: nf2 = NumberField(y^2 - 2, name='a', check=False)
sage: root2 = ANRoot(x^2 - 2, RIF(1, 2))
sage: gen2 = AlgebraicGenerator(nf2, root2)
sage: gen2
Number Field in a with defining polynomial y^2 - 2 with a in 1.
˓→414213562373095?
sage: nf3 = NumberField(y^2 - 3, name='a', check=False)
sage: root3 = ANRoot(x^2 - 3, RIF(1, 2))
sage: gen3 = AlgebraicGenerator(nf3, root3)
sage: gen3
Number Field in a with defining polynomial y^2 - 3 with a in 1.
˓→732050807568878?
sage: gen2.union(qq_generator) is gen2
True
sage: qq_generator.union(gen3) is gen3
True
sage: gen2.union(gen3)
Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 0.
˓→5176380902050415?
class sage.rings.qqbar. AlgebraicGeneratorRelation ( child1,
child1_poly,
child2_poly, parent)
Bases: sage.structure.sage_object.SageObject
child2,
A simple class for maintaining relations in the lattice of algebraic extensions.
class sage.rings.qqbar. AlgebraicNumber ( x)
Bases: sage.rings.qqbar.AlgebraicNumber_base
The class for algebraic numbers (complex numbers which are the roots of a polynomial with integer coefficients).
Much of its functionality is inherited from AlgebraicNumber_base .
complex_exact ( field)
Given a ComplexField , return the best possible approximation of this number in that field. Note that
if either component is sufficiently close to the halfway point between two floating-point numbers in the
corresponding RealField , then this will trigger exact computation, which may be very slow.
EXAMPLES:
sage: a = QQbar.zeta(9) + QQbar(I) + QQbar.zeta(9).conjugate(); a
1.532088886237957? + 1.000000000000000?*I
sage: a.complex_exact(CIF)
1.532088886237957? + 1*I
complex_number ( field)
Given the complex field field compute an accurate approximation of this element in that field.
The approximation will be off by at most two ulp’s in each component, except for components which are
very close to zero, which will have an absolute error at most 2−𝑝𝑟𝑒𝑐+1 where 𝑝𝑟𝑒𝑐 is the precision of the
field.
EXAMPLES:
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sage: a = QQbar.zeta(5)
sage: a.complex_number(CC)
0.309016994374947 + 0.951056516295154*I
sage: b = QQbar(2).sqrt() + QQbar(3).sqrt() * QQbar.gen()
sage: b.complex_number(ComplexField(128))
1.4142135623730950488016887242096980786 + 1.
˓→7320508075688772935274463415058723669*I
conjugate ( )
Returns the complex conjugate of self.
EXAMPLES:
sage: QQbar(3 + 4*I).conjugate()
3 - 4*I
sage: QQbar.zeta(7).conjugate()
0.6234898018587335? - 0.7818314824680299?*I
sage: QQbar.zeta(7) + QQbar.zeta(7).conjugate()
1.246979603717467? + 0.?e-18*I
imag ( )
Return the imaginary part of self.
EXAMPLES:
sage: QQbar.zeta(7).imag()
0.7818314824680299?
interval_exact ( field)
Given a ComplexIntervalField , compute the best possible approximation of this number in that
field. Note that if either the real or imaginary parts of this number are sufficiently close to some floatingpoint number (and, in particular, if either is exactly representable in floating-point), then this will trigger
exact computation, which may be very slow.
EXAMPLES:
sage: a = QQbar(I).sqrt(); a
0.7071067811865475? + 0.7071067811865475?*I
sage: a.interval_exact(CIF)
0.7071067811865475? + 0.7071067811865475?*I
sage: b = QQbar((1+I)*sqrt(2)/2)
sage: (a - b).interval(CIF)
0.?e-19 + 0.?e-18*I
sage: (a - b).interval_exact(CIF)
0
multiplicative_order ( )
Compute the multiplicative order of this algebraic real number. That is, find the smallest positive integer
𝑛 such that 𝑥𝑛 = 1. If there is no such 𝑛, returns +Infinity .
We first check that abs(x) is very close to 1. If so, we compute 𝑥 exactly and examine its argument.
EXAMPLES:
sage: QQbar(-sqrt(3)/2 - I/2).multiplicative_order()
12
sage: QQbar(1).multiplicative_order()
1
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sage: QQbar(-I).multiplicative_order()
4
sage: QQbar(707/1000 + 707/1000*I).multiplicative_order()
+Infinity
sage: QQbar(3/5 + 4/5*I).multiplicative_order()
+Infinity
norm ( )
Returns self * self.conjugate() . This is the algebraic definition of norm, if we view QQbar
as AA[I] .
EXAMPLES:
sage: QQbar(3 + 4*I).norm()
25
sage: type(QQbar(I).norm())
<class 'sage.rings.qqbar.AlgebraicReal'>
sage: QQbar.zeta(1007).norm()
1.000000000000000?
rational_argument ( )
Returns the argument of self, divided by 2𝜋, as long as this result is rational. Otherwise returns None.
Always triggers exact computation.
EXAMPLES:
sage:
1/8
sage:
1/3
sage:
-1/3
sage:
True
sage:
2/7
sage:
5/73
sage:
1/3
QQbar((1+I)*(sqrt(2)+sqrt(5))).rational_argument()
QQbar(-1 + I*sqrt(3)).rational_argument()
QQbar(-1 - I*sqrt(3)).rational_argument()
QQbar(3+4*I).rational_argument() is None
(QQbar(2)**(1/5) * QQbar.zeta(7)**2).rational_argument()
(QQbar.zeta(73)**5).rational_argument()
(QQbar.zeta(3)^65536).rational_argument()
real ( )
Return the real part of self.
EXAMPLES:
sage: QQbar.zeta(5).real()
0.3090169943749474?
class sage.rings.qqbar. AlgebraicNumber_base ( parent, x)
Bases: sage.structure.element.FieldElement
This is the common base class for algebraic numbers (complex numbers which are the zero of a polynomial in
Z[𝑥]) and algebraic reals (algebraic numbers which happen to be real).
AlgebraicNumber objects can be created using QQbar (== AlgebraicNumberField() ), and
AlgebraicReal objects can be created using AA (== AlgebraicRealField() ). They can be created
either by coercing a rational or a symbolic expression, or by using the QQbar.polynomial_root() or
AA.polynomial_root() method to construct a particular root of a polynomial with algebraic coefficients.
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Also, AlgebraicNumber and AlgebraicReal are closed under addition, subtraction, multiplication,
division (except by 0), and rational powers (including roots), except that for a negative AlgebraicReal ,
taking a power with an even denominator returns an AlgebraicNumber instead of an AlgebraicReal .
AlgebraicNumber and AlgebraicReal objects can be approximated to any desired precision. They
can be compared exactly; if the two numbers are very close, or are equal, this may require exact computation,
which can be extremely slow.
As long as exact computation is not triggered, computation with algebraic numbers should not be too much
slower than computation with intervals. As mentioned above, exact computation is triggered when comparing
two algebraic numbers which are very close together. This can be an explicit comparison in user code, but the
following list of actions (not necessarily complete) can also trigger exact computation:
•Dividing by an algebraic number which is very close to 0.
•Using an algebraic number which is very close to 0 as the leading coefficient in a polynomial.
•Taking a root of an algebraic number which is very close to 0.
The exact definition of “very close” is subject to change; currently, we compute our best approximation of the
two numbers using 128-bit arithmetic, and see if that’s sufficient to decide the comparison. Note that comparing
two algebraic numbers which are actually equal will always trigger exact computation, unless they are actually
the same object.
EXAMPLES:
sage: sqrt(QQbar(2))
1.414213562373095?
sage: sqrt(QQbar(2))^2 == 2
True
sage: x = polygen(QQbar)
sage: phi = QQbar.polynomial_root(x^2 - x - 1, RIF(1, 2))
sage: phi
1.618033988749895?
sage: phi^2 == phi+1
True
sage: AA(sqrt(65537))
256.0019531175495?
as_number_field_element ( minimal=False)
Returns a number field containing this value, a representation of this value as an element of that number
field, and a homomorphism from the number field back to AA or QQbar .
This may not return the smallest such number field, unless minimal=True is specified.
To compute a single number field containing multiple algebraic numbers, use the function
number_field_elements_from_algebraics instead.
EXAMPLES:
sage: QQbar(sqrt(8)).as_number_field_element()
(Number Field in a with defining polynomial y^2 - 2, 2*a, Ring morphism:
From: Number Field in a with defining polynomial y^2 - 2
To:
Algebraic Real Field
Defn: a |--> 1.414213562373095?)
sage: x = polygen(ZZ)
sage: p = x^3 + x^2 + x + 17
sage: (rt,) = p.roots(ring=AA, multiplicities=False); rt
-2.804642726932742?
sage: (nf, elt, hom) = rt.as_number_field_element()
sage: nf, elt, hom
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(Number Field in a with defining polynomial y^3 - 2*y^2 - 31*y - 50, a^2 ˓→5*a - 19, Ring morphism:
From: Number Field in a with defining polynomial y^3 - 2*y^2 - 31*y - 50
To:
Algebraic Real Field
Defn: a |--> 7.237653139801104?)
sage: hom(elt) == rt
True
We see an example where we do not get the minimal number field unless we specify minimal=True :
sage: rt2 = AA(sqrt(2))
sage: rt3 = AA(sqrt(3))
sage: rt3b = rt2 + rt3 - rt2
sage: rt3b.as_number_field_element()
(Number Field in a with defining polynomial y^4 - 4*y^2 + 1, -a^2 + 2, Ring
˓→morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To:
Algebraic Real Field
Defn: a |--> 0.5176380902050415?)
sage: rt3b.as_number_field_element(minimal=True)
(Number Field in a with defining polynomial y^2 - 3, a, Ring morphism:
From: Number Field in a with defining polynomial y^2 - 3
To:
Algebraic Real Field
Defn: a |--> 1.732050807568878?)
degree ( )
Return the degree of this algebraic number (the degree of its minimal polynomial, or equivalently, the
degree of the smallest algebraic extension of the rationals containing this number).
EXAMPLES:
sage:
1
sage:
2
sage:
5
sage:
2
QQbar(5/3).degree()
sqrt(QQbar(2)).degree()
QQbar(17).nth_root(5).degree()
sqrt(3+sqrt(QQbar(8))).degree()
exactify ( )
Compute an exact representation for this number.
EXAMPLES:
sage: two = QQbar(4).nth_root(4)^2
sage: two
2.000000000000000?
sage: two.exactify()
sage: two
2
interval ( field)
Given an interval field (real or complex, as appropriate) of precision 𝑝, compute an interval representation of self with diameter() at most 2−𝑝 ; then round that representation into the given field. Here
diameter() is relative diameter for intervals not containing 0, and absolute diameter for intervals that
do contain 0; thus, if the returned interval does not contain 0, it has at least 𝑝 − 1 good bits.
EXAMPLES:
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sage: RIF64 = RealIntervalField(64)
sage: x = AA(2).sqrt()
sage: y = x*x
sage: y = 1000 * y - 999 * y
sage: y.interval_fast(RIF64)
2.000000000000000?
sage: y.interval(RIF64)
2.000000000000000000?
sage: CIF64 = ComplexIntervalField(64)
sage: x = QQbar.zeta(11)
sage: x.interval_fast(CIF64)
0.8412535328311811689? + 0.5406408174555975821?*I
sage: x.interval(CIF64)
0.8412535328311811689? + 0.5406408174555975822?*I
The following implicitly use this method:
sage: RIF(AA(5).sqrt())
2.236067977499790?
sage: AA(-5).sqrt().interval(RIF)
Traceback (most recent call last):
...
TypeError: unable to convert 2.236067977499790?*I to real interval
interval_diameter ( diam)
Compute an interval representation of self with diameter() at most diam . The precision of the
returned value is unpredictable.
EXAMPLES:
sage: AA(2).sqrt().interval_diameter(1e-10)
1.4142135623730950488?
sage: AA(2).sqrt().interval_diameter(1e-30)
1.41421356237309504880168872420969807857?
sage: QQbar(2).sqrt().interval_diameter(1e-10)
1.4142135623730950488?
sage: QQbar(2).sqrt().interval_diameter(1e-30)
1.41421356237309504880168872420969807857?
interval_fast ( field)
Given a RealIntervalField or ComplexIntervalField , compute the value of this number
using interval arithmetic of at least the precision of the field, and return the value in that field. (More
precision may be used in the computation.) The returned interval may be arbitrarily imprecise, if this
number is the result of a sufficiently long computation chain.
EXAMPLES:
sage: x = AA(2).sqrt()
sage: x.interval_fast(RIF)
1.414213562373095?
sage: x.interval_fast(RealIntervalField(200))
1.414213562373095048801688724209698078569671875376948073176680?
sage: x = QQbar(I).sqrt()
sage: x.interval_fast(CIF)
0.7071067811865475? + 0.7071067811865475?*I
sage: x.interval_fast(RIF)
Traceback (most recent call last):
...
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TypeError: unable to convert 0.7071067811865475244? + 0.7071067811865475244?
˓→*I to real interval
is_integer ( )
Return True if this number is a integer
EXAMPLES:
sage: QQbar(2).is_integer()
True
sage: QQbar(1/2).is_integer()
False
is_square ( )
Return whether or not this number is square.
OUTPUT:
(boolean) True in all cases for elements of QQbar; True for non-negative elements of AA, otherwise False.
EXAMPLES:
sage:
True
sage:
False
sage:
True
sage:
True
AA(2).is_square()
AA(-2).is_square()
QQbar(-2).is_square()
QQbar(I).is_square()
minpoly ( )
Compute the minimal polynomial of this algebraic number. The minimal polynomial is the monic polynomial of least degree having this number as a root; it is unique.
EXAMPLES:
sage: QQbar(4).sqrt().minpoly()
x - 2
sage: ((QQbar(2).nth_root(4))^2).minpoly()
x^2 - 2
sage: v = sqrt(QQbar(2)) + sqrt(QQbar(3)); v
3.146264369941973?
sage: p = v.minpoly(); p
x^4 - 10*x^2 + 1
sage: p(RR(v.real()))
1.31006316905768e-14
nth_root ( n, all=False)
Return the n -th root of this number.
INPUT:
•all - bool (default: False ). If True , return a list of all 𝑛-th roots as complex algebraic numbers.
Warning: Note that for odd 𝑛, all=‘False‘ and negative real numbers, AlgebraicReal and
AlgebraicNumber values give different answers: AlgebraicReal values prefer real results,
and AlgebraicNumber values return the principal root.
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EXAMPLES:
sage: AA(-8).nth_root(3)
-2
sage: QQbar(-8).nth_root(3)
1.000000000000000? + 1.732050807568878?*I
sage: QQbar.zeta(12).nth_root(15)
0.9993908270190957? + 0.03489949670250097?*I
You can get all n -th roots of algebraic numbers:
sage: AA(-8).nth_root(3, all=True)
[1.000000000000000? + 1.732050807568878?*I,
-2.000000000000000? + 0.?e-18*I,
1.000000000000000? - 1.732050807568878?*I]
sage: QQbar(1+I).nth_root(4, all=True)
[1.069553932363986? + 0.2127475047267431?*I,
-0.2127475047267431? + 1.069553932363986?*I,
-1.069553932363986? - 0.2127475047267431?*I,
0.2127475047267431? - 1.069553932363986?*I]
radical_expression ( )
Attempt to obtain a symbolic expression using radicals. If no exact symbolic expression can be found, the
algebraic number will be returned without modification.
EXAMPLES:
sage: AA(1/sqrt(5)).radical_expression()
1/5*sqrt(5)
sage: AA(sqrt(5 + sqrt(5))).radical_expression()
sqrt(sqrt(5) + 5)
sage: QQbar.zeta(5).radical_expression()
1/4*sqrt(5) + 1/2*sqrt(-1/2*sqrt(5) - 5/2) - 1/4
sage: a = QQ[x](x^7 - x - 1).roots(AA, False)[0]
sage: a.radical_expression()
1.112775684278706?
sage: a.radical_expression().parent() == SR
False
sage: a = sorted(QQ[x](x^7-x-1).roots(QQbar, False), key=imag)[0]
sage: a.radical_expression()
-0.3636235193291805? - 0.9525611952610331?*I
sage: QQbar.zeta(5).imag().radical_expression()
1/2*sqrt(1/2*sqrt(5) + 5/2)
sage: AA(5/3).radical_expression()
5/3
sage: AA(5/3).radical_expression().parent() == SR
True
sage: QQbar(0).radical_expression()
0
sage: a = AA(sqrt(2) + 10^25)
sage: p = a.minpoly()
sage: v = a._value
sage: f = ComplexIntervalField(v.prec())
sage: [f(b.rhs()).overlaps(f(v)) for b in SR(p).solve(x)]
[True, True]
sage: a.radical_expression()
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sqrt(2) + 10000000000000000000000000
simplify ( )
Compute an exact representation for this number, in the smallest possible number field.
EXAMPLES:
sage: rt2 = AA(sqrt(2))
sage: rt3 = AA(sqrt(3))
sage: rt2b = rt3 + rt2 - rt3
sage: rt2b.exactify()
sage: rt2b._exact_value()
a^3 - 3*a where a^4 - 4*a^2 + 1 = 0 and a in 1.931851652578137?
sage: rt2b.simplify()
sage: rt2b._exact_value()
a where a^2 - 2 = 0 and a in 1.414213562373095?
sqrt ( all=False, extend=True)
Return the square root(s) of this number.
INPUT:
•extend - bool (default: True); ignored if self is in QQbar, or positive in AA. If self is negative in
AA, do the following: if True, return a square root of self in QQbar, otherwise raise a ValueError.
•all - bool (default: False); if True, return a list of all square roots. If False, return just one square
root, or raise an ValueError if self is a negative element of AA and extend=False.
OUTPUT:
Either the principal square root of self, or a list of its square roots (with the principal one first).
EXAMPLES:
sage: AA(2).sqrt()
1.414213562373095?
sage: QQbar(I).sqrt()
0.7071067811865475? + 0.7071067811865475?*I
sage: QQbar(I).sqrt(all=True)
[0.7071067811865475? + 0.7071067811865475?*I, -0.7071067811865475? - 0.
˓→7071067811865475?*I]
sage: a = QQbar(0)
sage: a.sqrt()
0
sage: a.sqrt(all=True)
[0]
sage: a = AA(0)
sage: a.sqrt()
0
sage: a.sqrt(all=True)
[0]
This second example just shows that the program doesn’t care where 0 is defined, it gives the same answer
regardless. After all, how many ways can you square-root zero?
sage: AA(-2).sqrt()
1.414213562373095?*I
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sage: AA(-2).sqrt(all=True)
[1.414213562373095?*I, -1.414213562373095?*I]
sage: AA(-2).sqrt(extend=False)
Traceback (most recent call last):
...
ValueError: -2 is not a square in AA, being negative. Use extend = True for a
˓→square root in QQbar.
class sage.rings.qqbar. AlgebraicPolynomialTracker ( poly)
Bases: sage.structure.sage_object.SageObject
Keeps track of a polynomial used for algebraic numbers.
If multiple algebraic numbers are created as roots of a single polynomial, this allows the polynomial and information about the polynomial to be shared. This reduces work if the polynomial must be recomputed at higher
precision, or if it must be factored.
This class is private, and should only be constructed by AA.common_polynomial()
or
QQbar.common_polynomial() , and should only be used as an argument to AA.polynomial_root()
or QQbar.polynomial_root() . (It doesn’t matter whether you create the common polynomial with
AA.common_polynomial() or QQbar.common_polynomial() .)
EXAMPLES:
sage: x = polygen(QQbar)
sage: P = QQbar.common_polynomial(x^2 - x - 1)
sage: P
x^2 - x - 1
sage: QQbar.polynomial_root(P, RIF(1, 2))
1.618033988749895?
complex_roots ( prec, multiplicity)
Find the roots of self in the complex field to precision prec.
EXAMPLES:
sage: x = polygen(ZZ)
sage: cp = AA.common_polynomial(x^4 - 2)
Note that the precision is not guaranteed to find the tightest possible interval since complex_roots() depends
on the underlying BLAS implementation.
sage: cp.complex_roots(30, 1)
[-1.18920711500272...?,
1.189207115002721?,
-1.189207115002721?*I,
1.189207115002721?*I]
exactify ( )
Compute a common field that holds all of the algebraic coefficients of this polynomial, then factor the
polynomial over that field. Store the factors for later use (ignoring multiplicity).
EXAMPLES:
sage: x = polygen(AA)
sage: p = sqrt(AA(2)) * x^2 - sqrt(AA(3))
sage: cp = AA.common_polynomial(p)
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sage: cp._exact
False
sage: cp.exactify()
sage: cp._exact
True
factors ( )
EXAMPLES:
sage: x=polygen(QQ); f=QQbar.common_polynomial(x^4 + 4)
sage: f.factors()
[y^2 - 2*y + 2, y^2 + 2*y + 2]
generator ( )
Return an AlgebraicGenerator for a number field containing all the coefficients of self.
EXAMPLES:
sage: x = polygen(AA)
sage: p = sqrt(AA(2)) * x^2 - sqrt(AA(3))
sage: cp = AA.common_polynomial(p)
sage: cp.generator()
Number Field in a with defining polynomial y^4 - 4*y^2 + 1 with a in 1.
˓→931851652578137?
is_complex ( )
Return True if the coefficients of this polynomial are non-real.
EXAMPLES:
sage:
sage:
sage:
False
sage:
True
x = polygen(QQ); f = x^3 - 7
g = AA.common_polynomial(f)
g.is_complex()
QQbar.common_polynomial(x^3 - QQbar(I)).is_complex()
poly ( )
Return the underlying polynomial of self.
EXAMPLES:
sage: x = polygen(QQ); f = x^3 - 7
sage: g = AA.common_polynomial(f)
sage: g.poly() == f
True
class sage.rings.qqbar. AlgebraicReal ( x)
Bases: sage.rings.qqbar.AlgebraicNumber_base
Create an algebraic real from x, possibly taking the real part of x.
ceil ( )
Return the smallest integer not smaller than self .
EXAMPLES:
sage: AA(sqrt(2)).ceil()
2
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sage: AA(-sqrt(2)).ceil()
-1
sage: AA(42).ceil()
42
conjugate ( )
Returns the complex conjugate of self, i.e. returns itself.
EXAMPLES:
sage: a = AA(sqrt(2) + sqrt(3))
sage: a.conjugate()
3.146264369941973?
sage: a.conjugate() is a
True
floor ( )
Return the largest integer not greater than self .
EXAMPLES:
sage: AA(sqrt(2)).floor()
1
sage: AA(-sqrt(2)).floor()
-2
sage: AA(42).floor()
42
imag ( )
Returns the imaginary part of this algebraic real (so it always returns 0).
EXAMPLES:
sage: a = AA(sqrt(2) + sqrt(3))
sage: a.imag()
0
sage: parent(a.imag())
Algebraic Real Field
interval_exact ( field)
Given a RealIntervalField , compute the best possible approximation of this number in that field.
Note that if this number is sufficiently close to some floating-point number (and, in particular, if this
number is exactly representable in floating-point), then this will trigger exact computation, which may be
very slow.
EXAMPLES:
sage: x = AA(2).sqrt()
sage: y = x*x
sage: x.interval(RIF)
1.414213562373095?
sage: x.interval_exact(RIF)
1.414213562373095?
sage: y.interval(RIF)
2.000000000000000?
sage: y.interval_exact(RIF)
2
sage: z = 1 + AA(2).sqrt() / 2^200
sage: z.interval(RIF)
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1.000000000000001?
sage: z.interval_exact(RIF)
1.000000000000001?
real ( )
Returns the real part of this algebraic real (so it always returns self).
EXAMPLES:
sage: a = AA(sqrt(2) + sqrt(3))
sage: a.real()
3.146264369941973?
sage: a.real() is a
True
real_exact ( field)
Given a RealField , compute the best possible approximation of this number in that field. Note that if
this number is sufficiently close to the halfway point between two floating-point numbers in the field (for
the default round-to-nearest mode) or if the number is sufficiently close to a floating-point number in the
field (for directed rounding modes), then this will trigger exact computation, which may be very slow.
The rounding mode of the field is respected.
EXAMPLES:
sage: x = AA(2).sqrt()^2
sage: x.real_exact(RR)
2.00000000000000
sage: x.real_exact(RealField(53, rnd='RNDD'))
2.00000000000000
sage: x.real_exact(RealField(53, rnd='RNDU'))
2.00000000000000
sage: x.real_exact(RealField(53, rnd='RNDZ'))
2.00000000000000
sage: (-x).real_exact(RR)
-2.00000000000000
sage: (-x).real_exact(RealField(53, rnd='RNDD'))
-2.00000000000000
sage: (-x).real_exact(RealField(53, rnd='RNDU'))
-2.00000000000000
sage: (-x).real_exact(RealField(53, rnd='RNDZ'))
-2.00000000000000
sage: y = (x-2).real_exact(RR).abs()
sage: y == 0.0 or y == -0.0 # the sign of 0.0 is not significant in MPFI
True
sage: y = (x-2).real_exact(RealField(53, rnd='RNDD'))
sage: y == 0.0 or y == -0.0 # same as above
True
sage: y = (x-2).real_exact(RealField(53, rnd='RNDU'))
sage: y == 0.0 or y == -0.0 # idem
True
sage: y = (x-2).real_exact(RealField(53, rnd='RNDZ'))
sage: y == 0.0 or y == -0.0 # ibidem
True
sage: y = AA(2).sqrt()
sage: y.real_exact(RR)
1.41421356237310
sage: y.real_exact(RealField(53, rnd='RNDD'))
1.41421356237309
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sage: y.real_exact(RealField(53, rnd='RNDU'))
1.41421356237310
sage: y.real_exact(RealField(53, rnd='RNDZ'))
1.41421356237309
real_number ( field)
Given a RealField , compute a good approximation to self in that field. The approximation will be off
by at most two ulp’s, except for numbers which are very close to 0, which will have an absolute error at
most 2**(-(field.prec()-1)) . Also, the rounding mode of the field is respected.
EXAMPLES:
sage: x = AA(2).sqrt()^2
sage: x.real_number(RR)
2.00000000000000
sage: x.real_number(RealField(53, rnd='RNDD'))
1.99999999999999
sage: x.real_number(RealField(53, rnd='RNDU'))
2.00000000000001
sage: x.real_number(RealField(53, rnd='RNDZ'))
1.99999999999999
sage: (-x).real_number(RR)
-2.00000000000000
sage: (-x).real_number(RealField(53, rnd='RNDD'))
-2.00000000000001
sage: (-x).real_number(RealField(53, rnd='RNDU'))
-1.99999999999999
sage: (-x).real_number(RealField(53, rnd='RNDZ'))
-1.99999999999999
sage: (x-2).real_number(RR)
5.42101086242752e-20
sage: (x-2).real_number(RealField(53, rnd='RNDD'))
-1.08420217248551e-19
sage: (x-2).real_number(RealField(53, rnd='RNDU'))
2.16840434497101e-19
sage: (x-2).real_number(RealField(53, rnd='RNDZ'))
0.000000000000000
sage: y = AA(2).sqrt()
sage: y.real_number(RR)
1.41421356237309
sage: y.real_number(RealField(53, rnd='RNDD'))
1.41421356237309
sage: y.real_number(RealField(53, rnd='RNDU'))
1.41421356237310
sage: y.real_number(RealField(53, rnd='RNDZ'))
1.41421356237309
round ( )
Round self to the nearest integer.
EXAMPLES:
sage: AA(sqrt(2)).round()
1
sage: AA(1/2).round()
1
sage: AA(-1/2).round()
-1
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sign ( )
Compute the sign of this algebraic number (return -1 if negative, 0 if zero, or 1 if positive).
Computes an interval enclosing this number using 128-bit interval arithmetic; if this interval includes 0,
then fall back to exact computation (which can be very slow).
EXAMPLES:
sage: AA(-5).nth_root(7).sign()
-1
sage: (AA(2).sqrt() - AA(2).sqrt()).sign()
0
trunc ( )
Round self to the nearest integer toward zero.
EXAMPLES:
sage:
1
sage:
-1
sage:
1
sage:
-1
AA(sqrt(2)).trunc()
AA(-sqrt(2)).trunc()
AA(1).trunc()
AA(-1).trunc()
class sage.rings.qqbar. AlgebraicRealField
Bases: sage.misc.fast_methods.Singleton , sage.rings.qqbar.AlgebraicField_common
The field of algebraic reals.
algebraic_closure ( )
Return the algebraic closure of this field, which is the field Q of algebraic numbers.
EXAMPLES:
sage: AA.algebraic_closure()
Algebraic Field
completion ( p, prec, extras={})
Return the completion of self at the place 𝑝. Only implemented for 𝑝 = ∞ at present.
INPUT:
•p – either a prime (not implemented at present) or Infinity
•prec – precision of approximate field to return
•extras – a dict of extra keyword arguments for the RealField constructor
EXAMPLES:
sage: AA.completion(infinity, 500)
Real Field with 500 bits of precision
sage: AA.completion(infinity, prec=53, extras={'type':'RDF'})
Real Double Field
sage: AA.completion(infinity, 53) is RR
True
sage: AA.completion(7, 10)
Traceback (most recent call last):
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...
NotImplementedError
gen ( n=0)
Return the 𝑛-th element of the tuple returned by gens() .
EXAMPLES:
sage: AA.gen(0)
1
sage: AA.gen(1)
Traceback (most recent call last):
...
IndexError: n must be 0
gens ( )
Return a set of generators for this field. As this field is not finitely generated, we opt for just returning 1.
EXAMPLES:
sage: AA.gens()
(1,)
ngens ( )
Return the size of the tuple returned by gens() .
EXAMPLES:
sage: AA.ngens()
1
polynomial_root ( poly, interval, multiplicity=1)
Given a polynomial with algebraic coefficients and an interval enclosing exactly one root of the polynomial, constructs an algebraic real representation of that root.
The polynomial need not be irreducible, or even squarefree; but if the given root is a multiple root, its
multiplicity must be specified. (IMPORTANT NOTE: Currently, multiplicity-𝑘 roots are handled by taking
the (𝑘 − 1)-st derivative of the polynomial. This means that the interval must enclose exactly one root of
this derivative.)
The conditions on the arguments (that the interval encloses exactly one root, and that multiple roots match
the given multiplicity) are not checked; if they are not satisfied, an error may be thrown (possibly later,
when the algebraic number is used), or wrong answers may result.
Note that if you are constructing multiple roots of a single polynomial, it is better to use
AA.common_polynomial (or QQbar.common_polynomial ; the two are equivalent) to get a
shared polynomial.
EXAMPLES:
sage: x = polygen(AA)
sage: phi = AA.polynomial_root(x^2 - x - 1, RIF(1, 2)); phi
1.618033988749895?
sage: p = (x-1)^7 * (x-2)
sage: r = AA.polynomial_root(p, RIF(9/10, 11/10), multiplicity=7)
sage: r; r == 1
1.000000000000000?
True
sage: p = (x-phi)*(x-sqrt(AA(2)))
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sage: r = AA.polynomial_root(p, RIF(1, 3/2))
sage: r; r == sqrt(AA(2))
1.414213562373095?
True
We allow complex polynomials, as long as the particular root in question is real.
sage: K.<im> = QQ[I]
sage: x = polygen(K)
sage: p = (im + 1) * (x^3 - 2); p
(I + 1)*x^3 - 2*I - 2
sage: r = AA.polynomial_root(p, RIF(1, 2)); r^3
2.000000000000000?
zeta ( n=2)
Return an 𝑛-th root of unity in this field. This will raise a ValueError if 𝑛 ̸= {1, 2} since no such root
exists.
INPUT:
•n (integer) – default 2
EXAMPLES:
sage: AA.zeta(1)
1
sage: AA.zeta(2)
-1
sage: AA.zeta()
-1
sage: AA.zeta(3)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in algebraic reals
Some silly inputs:
sage: AA.zeta(Mod(-5, 7))
-1
sage: AA.zeta(0)
Traceback (most recent call last):
...
ValueError: no n-th root of unity in algebraic reals
sage.rings.qqbar. an_binop_element ( a, b, op)
Add, subtract, multiply or divide two elements represented as elements of number fields.
EXAMPLES:
sage: sqrt2 = QQbar(2).sqrt()
sage: sqrt3 = QQbar(3).sqrt()
sage: sqrt5 = QQbar(5).sqrt()
sage: a = sqrt2 + sqrt3; a.exactify()
sage: b = sqrt3 + sqrt5; b.exactify()
sage: type(a._descr)
<class 'sage.rings.qqbar.ANExtensionElement'>
sage: from sage.rings.qqbar import an_binop_element
sage: an_binop_element(a, b, operator.add)
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<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: an_binop_element(a, b, operator.sub)
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: an_binop_element(a, b, operator.mul)
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: an_binop_element(a, b, operator.div)
<class 'sage.rings.qqbar.ANBinaryExpr'>
The code tries to use existing unions of number fields:
sage: sqrt17 = QQbar(17).sqrt()
sage: sqrt19 = QQbar(19).sqrt()
sage: a = sqrt17 + sqrt19
sage: b = sqrt17 * sqrt19 - sqrt17 + sqrt19 * (sqrt17 + 2)
sage: b, type(b._descr)
(40.53909377268655?, <class 'sage.rings.qqbar.ANBinaryExpr'>)
sage: a.exactify()
sage: b = sqrt17 * sqrt19 - sqrt17 + sqrt19 * (sqrt17 + 2)
sage: b, type(b._descr)
(40.53909377268655?, <class 'sage.rings.qqbar.ANExtensionElement'>)
sage.rings.qqbar. an_binop_expr ( a, b, op)
Add, subtract, multiply or divide algebraic numbers represented as binary expressions.
INPUT:
•a , b – two elements
•op – an operator
EXAMPLES:
sage: a = QQbar(sqrt(2)) + QQbar(sqrt(3))
sage: b = QQbar(sqrt(3)) + QQbar(sqrt(5))
sage: type(a._descr); type(b._descr)
<class 'sage.rings.qqbar.ANBinaryExpr'>
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: from sage.rings.qqbar import an_binop_expr
sage: x = an_binop_expr(a, b, operator.add); x
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: x.exactify()
-6/7*a^7 + 2/7*a^6 + 71/7*a^5 - 26/7*a^4 - 125/7*a^3 + 72/7*a^2 + 43/7*a - 47/7
˓→where a^8 - 12*a^6 + 23*a^4 - 12*a^2 + 1 = 0 and a in 3.12580...?
sage: a = QQbar(sqrt(2)) + QQbar(sqrt(3))
sage: b = QQbar(sqrt(3)) + QQbar(sqrt(5))
sage: type(a._descr)
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: x = an_binop_expr(a, b, operator.mul); x
<class 'sage.rings.qqbar.ANBinaryExpr'>
sage: x.exactify()
2*a^7 - a^6 - 24*a^5 + 12*a^4 + 46*a^3 - 22*a^2 - 22*a + 9 where a^8 - 12*a^6 +
˓→23*a^4 - 12*a^2 + 1 = 0 and a in 3.1258...?
sage.rings.qqbar. an_binop_rational ( a, b, op)
Used to add, subtract, multiply or divide algebraic numbers.
Used when both are actually rational.
EXAMPLES:
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sage: from sage.rings.qqbar import an_binop_rational
sage: f = an_binop_rational(QQbar(2), QQbar(3/7), operator.add)
sage: f
17/7
sage: type(f)
<class 'sage.rings.qqbar.ANRational'>
sage: f = an_binop_rational(QQbar(2), QQbar(3/7), operator.mul)
sage: f
6/7
sage: type(f)
<class 'sage.rings.qqbar.ANRational'>
sage.rings.qqbar. clear_denominators ( poly)
Takes a monic polynomial and rescales the variable to get a monic polynomial with “integral” coefficients.
Works on any univariate polynomial whose base ring has a denominator() method that returns integers;
for example, the base ring might be Q or a number field.
Returns the scale factor and the new polynomial.
(Inspired by Pari’s primitive_pol_to_monic() .)
We assume that coefficient denominators are “small”; the algorithm factors the denominators, to give the smallest possible scale factor.
EXAMPLES:
sage: from sage.rings.qqbar import clear_denominators
sage: _.<x> = QQ['x']
sage: clear_denominators(x + 3/2)
(2, x + 3)
sage: clear_denominators(x^2 + x/2 + 1/4)
(2, x^2 + x + 1)
sage.rings.qqbar. conjugate_expand ( v)
If the interval v (which may be real or complex) includes some purely real numbers, return v' containing v such that v' == v'.conjugate() . Otherwise return v unchanged. (Note that if v' ==
v'.conjugate() , and v' includes one non-real root of a real polynomial, then v' also includes the
conjugate of that root. Also note that the diameter of the return value is at most twice the diameter of the input.)
EXAMPLES:
sage: from sage.rings.qqbar import conjugate_expand
sage: conjugate_expand(CIF(RIF(0, 1), RIF(1, 2))).str(style='brackets')
'[0.00000000000000000 .. 1.0000000000000000] + [1.0000000000000000 .. 2.
˓→0000000000000000]*I'
sage: conjugate_expand(CIF(RIF(0, 1), RIF(0, 1))).str(style='brackets')
'[0.00000000000000000 .. 1.0000000000000000] + [-1.0000000000000000 .. 1.
˓→0000000000000000]*I'
sage: conjugate_expand(CIF(RIF(0, 1), RIF(-2, 1))).str(style='brackets')
'[0.00000000000000000 .. 1.0000000000000000] + [-2.0000000000000000 .. 2.
˓→0000000000000000]*I'
sage: conjugate_expand(RIF(1, 2)).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
sage.rings.qqbar. conjugate_shrink ( v)
If the interval v includes some purely real numbers, return a real interval containing only those real numbers.
Otherwise return v unchanged.
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If v includes exactly one root of a real polynomial, and v was returned by conjugate_expand() , then
conjugate_shrink(v) still includes that root, and is a RealIntervalFieldElement iff the root in
question is real.
EXAMPLES:
sage: from sage.rings.qqbar import conjugate_shrink
sage: conjugate_shrink(RIF(3, 4)).str(style='brackets')
'[3.0000000000000000 .. 4.0000000000000000]'
sage: conjugate_shrink(CIF(RIF(1, 2), RIF(1, 2))).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000] + [1.0000000000000000 .. 2.
˓→0000000000000000]*I'
sage: conjugate_shrink(CIF(RIF(1, 2), RIF(0, 1))).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
sage: conjugate_shrink(CIF(RIF(1, 2), RIF(-1, 2))).str(style='brackets')
'[1.0000000000000000 .. 2.0000000000000000]'
sage.rings.qqbar. do_polred ( poly)
Find a polynomial of reasonably small discriminant that generates the same number field as poly , using the
PARI polredbest function.
INPUT:
•poly - a monic irreducible polynomial with integer coefficients.
OUTPUT:
A triple (elt_fwd , elt_back , new_poly ), where:
•new_poly is the new polynomial defining the same number field,
•elt_fwd is a polynomial expression for a root of the new polynomial in terms of a root of the original
polynomial,
•elt_back is a polynomial expression for a root of the original polynomial in terms of a root of the new
polynomial.
EXAMPLES:
sage: from sage.rings.qqbar import do_polred
sage: R.<x> = QQ['x']
sage: oldpol = x^2 - 5
sage: fwd, back, newpol = do_polred(oldpol)
sage: newpol
x^2 - x - 1
sage: Kold.<a> = NumberField(oldpol)
sage: Knew.<b> = NumberField(newpol)
sage: newpol(fwd(a))
0
sage: oldpol(back(b))
0
sage: do_polred(x^2 - x - 11)
(1/3*x + 1/3, 3*x - 1, x^2 - x - 1)
sage: do_polred(x^3 + 123456)
(1/4*x, 4*x, x^3 + 1929)
This shows that trac ticket #13054 has been fixed:
sage: do_polred(x^4 - 4294967296*x^2 + 54265257667816538374400)
(1/4*x, 4*x, x^4 - 268435456*x^2 + 211973662764908353025)
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sage.rings.qqbar. find_zero_result ( fn, l)
l is a list of some sort. fn is a function which maps an element of l and a precision into an interval (either real
or complex) of that precision, such that for sufficient precision, exactly one element of l results in an interval
containing 0. Returns that one element of l .
EXAMPLES:
sage: from sage.rings.qqbar import find_zero_result
sage: _.<x> = QQ['x']
sage: delta = 10^(-70)
sage: p1 = x - 1
sage: p2 = x - 1 - delta
sage: p3 = x - 1 + delta
sage: p2 == find_zero_result(lambda p, prec: p(RealIntervalField(prec)(1 +
˓→delta)), [p1, p2, p3])
True
sage.rings.qqbar. get_AA_golden_ratio ( )
Return the golden ratio as an element of the algebraic
sage.symbolic.constants.golden_ratio._algebraic_() .
real
field.
Used
by
EXAMPLES:
sage: AA(golden_ratio) # indirect doctest
1.618033988749895?
sage.rings.qqbar. is_AlgebraicField ( F)
Check whether F is an AlgebraicField instance.
EXAMPLES:
sage: from sage.rings.qqbar import is_AlgebraicField
sage: [is_AlgebraicField(x) for x in [AA, QQbar, None, 0, "spam"]]
[False, True, False, False, False]
sage.rings.qqbar. is_AlgebraicField_common ( F)
Check whether F is an AlgebraicField_common instance.
EXAMPLES:
sage: from sage.rings.qqbar import is_AlgebraicField_common
sage: [is_AlgebraicField_common(x) for x in [AA, QQbar, None, 0, "spam"]]
[True, True, False, False, False]
sage.rings.qqbar. is_AlgebraicNumber ( x)
Test if x is an instance of AlgebraicNumber . For internal use.
EXAMPLES:
sage:
sage:
False
sage:
True
sage:
False
from sage.rings.qqbar import is_AlgebraicNumber
is_AlgebraicNumber(AA(sqrt(2)))
is_AlgebraicNumber(QQbar(sqrt(2)))
is_AlgebraicNumber("spam")
sage.rings.qqbar. is_AlgebraicReal ( x)
Test if x is an instance of AlgebraicReal . For internal use.
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EXAMPLES:
sage:
sage:
True
sage:
False
sage:
False
from sage.rings.qqbar import is_AlgebraicReal
is_AlgebraicReal(AA(sqrt(2)))
is_AlgebraicReal(QQbar(sqrt(2)))
is_AlgebraicReal("spam")
sage.rings.qqbar. is_AlgebraicRealField ( F)
Check whether F is an AlgebraicRealField instance. For internal use.
EXAMPLES:
sage: from sage.rings.qqbar import is_AlgebraicRealField
sage: [is_AlgebraicRealField(x) for x in [AA, QQbar, None, 0, "spam"]]
[True, False, False, False, False]
sage.rings.qqbar. isolating_interval ( intv_fn, pol)
intv_fn is a function that takes a precision and returns an interval of that precision containing some particular
root of pol. (It must return better approximations as the precision increases.) pol is an irreducible polynomial
with rational coefficients.
Returns an interval containing at most one root of pol.
EXAMPLES:
sage: from sage.rings.qqbar import isolating_interval
sage: _.<x> = QQ['x']
sage: isolating_interval(lambda prec: sqrt(RealIntervalField(prec)(2)), x^2 - 2)
1.4142135623730950488?
And an example that requires more precision:
sage: delta = 10^(-70)
sage: p = (x - 1) * (x - 1 - delta) * (x - 1 + delta)
sage: isolating_interval(lambda prec: RealIntervalField(prec)(1 + delta), p)
1.
˓→0000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000
˓→
The function also works with complex intervals and complex roots:
sage: p = x^2 - x + 13/36
sage: isolating_interval(lambda prec: ComplexIntervalField(prec)(1/2, 1/3), p)
0.500000000000000000000? + 0.3333333333333333334?*I
sage.rings.qqbar. number_field_elements_from_algebraics ( numbers, minimal=False)
Given a sequence of elements of either AA or QQbar (or a mixture), computes a number field containing all
of these elements, these elements as members of that number field, and a homomorphism from the number field
back to AA or QQbar .
This may not return the smallest such number field, unless minimal=True is specified.
Also, a single number can be passed, rather than a sequence; and any values which are not elements of AA or
QQbar will automatically be coerced to QQbar
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This function may be useful for efficiency reasons: doing exact computations in the corresponding number field
will be faster than doing exact computations directly in AA or QQbar .
EXAMPLES:
We can use this to compute the splitting field of a polynomial. (Unfortunately this takes an unreasonably long
time for non-toy examples.):
sage: x = polygen(QQ)
sage: p = x^3 + x^2 + x + 17
sage: rts = p.roots(ring=QQbar, multiplicities=False)
sage: splitting = number_field_elements_from_algebraics(rts)[0]; splitting
Number Field in a with defining polynomial y^6 - 40*y^4 - 22*y^3 + 873*y^2 +
˓→1386*y + 594
sage: p.roots(ring=splitting)
[(361/29286*a^5 - 19/3254*a^4 - 14359/29286*a^3 + 401/29286*a^2 + 18183/1627*a +
˓→15930/1627, 1), (49/117144*a^5 - 179/39048*a^4 - 3247/117144*a^3 + 22553/
˓→117144*a^2 + 1744/4881*a - 17195/6508, 1), (-1493/117144*a^5 + 407/39048*a^4 +
˓→60683/117144*a^3 - 24157/117144*a^2 - 56293/4881*a - 53033/6508, 1)]
sage: rt2 = AA(sqrt(2)); rt2
1.414213562373095?
sage: rt3 = AA(sqrt(3)); rt3
1.732050807568878?
sage: qqI = QQbar.zeta(4); qqI
I
sage: z3 = QQbar.zeta(3); z3
-0.500000000000000? + 0.866025403784439?*I
sage: rt2b = rt3 + rt2 - rt3; rt2b
1.414213562373095?
sage: rt2c = z3 + rt2 - z3; rt2c
1.414213562373095? + 0.?e-19*I
sage: number_field_elements_from_algebraics(rt2)
(Number Field in a with defining polynomial y^2 - 2, a, Ring morphism:
From: Number Field in a with defining polynomial y^2 - 2
To:
Algebraic Real Field
Defn: a |--> 1.414213562373095?)
sage: number_field_elements_from_algebraics((rt2,rt3))
(Number Field in a with defining polynomial y^4 - 4*y^2 + 1, [-a^3 + 3*a, -a^2 +
˓→2], Ring morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To:
Algebraic Real Field
Defn: a |--> 0.5176380902050415?)
We’ve created rt2b in such a way that sage doesn’t initially know that it’s in a degree-2 extension of Q:
sage: number_field_elements_from_algebraics(rt2b)
(Number Field in a with defining polynomial y^4 - 4*y^2 + 1, -a^3 + 3*a, Ring
˓→morphism:
From: Number Field in a with defining polynomial y^4 - 4*y^2 + 1
To:
Algebraic Real Field
Defn: a |--> 0.5176380902050415?)
We can specify minimal=True if we want the smallest number field:
sage: number_field_elements_from_algebraics(rt2b, minimal=True)
(Number Field in a with defining polynomial y^2 - 2, a, Ring morphism:
From: Number Field in a with defining polynomial y^2 - 2
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To:
Algebraic Real Field
Defn: a |--> 1.414213562373095?)
Things work fine with rational numbers, too:
sage: number_field_elements_from_algebraics((QQbar(1/2), AA(17)))
(Rational Field, [1/2, 17], Ring morphism:
From: Rational Field
To:
Algebraic Real Field
Defn: 1 |--> 1)
Or we can just pass in symbolic expressions, as long as they can be coerced into QQbar :
sage: number_field_elements_from_algebraics((sqrt(7), sqrt(9), sqrt(11)))
(Number Field in a with defining polynomial y^4 - 9*y^2 + 1, [-a^3 + 8*a, 3, -a^3
˓→+ 10*a], Ring morphism:
From: Number Field in a with defining polynomial y^4 - 9*y^2 + 1
To:
Algebraic Real Field
Defn: a |--> 0.3354367396454047?)
Here we see an example of doing some computations with number field elements, and then mapping them back
into QQbar :
sage: (fld,nums,hom) = number_field_elements_from_algebraics((rt2, rt3, qqI, z3))
sage: fld,nums,hom # random
(Number Field in a with defining polynomial y^8 - y^4 + 1, [-a^5 + a^3 + a, a^6 ˓→2*a^2, a^6, -a^4], Ring morphism:
From: Number Field in a with defining polynomial y^8 - y^4 + 1
To:
Algebraic Field
Defn: a |--> -0.2588190451025208? - 0.9659258262890683?*I)
sage: (nfrt2, nfrt3, nfI, nfz3) = nums
sage: hom(nfrt2)
1.414213562373095? + 0.?e-18*I
sage: nfrt2^2
2
sage: nfrt3^2
3
sage: nfz3 + nfz3^2
-1
sage: nfI^2
-1
sage: sum = nfrt2 + nfrt3 + nfI + nfz3; sum
2*a^6 + a^5 - a^4 - a^3 - 2*a^2 - a
sage: hom(sum)
2.646264369941973? + 1.866025403784439?*I
sage: hom(sum) == rt2 + rt3 + qqI + z3
True
sage: [hom(n) for n in nums] == [rt2, rt3, qqI, z3]
True
sage.rings.qqbar. prec_seq ( )
Return a generator object which iterates over an infinite increasing sequence of precisions to be tried in various
numerical computations.
Currently just returns powers of 2 starting at 64.
EXAMPLES:
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sage: g = sage.rings.qqbar.prec_seq()
sage: [next(g), next(g), next(g)]
[64, 128, 256]
sage.rings.qqbar. rational_exact_root ( r, d)
Checks whether the rational 𝑟 is an exact 𝑑‘th power. If so, returns the 𝑑‘th root of 𝑟; otherwise, returns None.
EXAMPLES:
sage: from sage.rings.qqbar import rational_exact_root
sage: rational_exact_root(16/81, 4)
2/3
sage: rational_exact_root(8/81, 3) is None
True
sage.rings.qqbar. short_prec_seq ( )
Return a sequence of precisions to try in cases when an infinite-precision computation is possible: returns a
couple of small powers of 2 and then None .
EXAMPLES:
sage: from sage.rings.qqbar import short_prec_seq
sage: short_prec_seq()
(64, 128, None)
sage.rings.qqbar. t1
alias of ANBinaryExpr
sage.rings.qqbar. t2
alias of ANRoot
sage.rings.qqbar. tail_prec_seq ( )
A generator over precisions larger than those in short_prec_seq() .
EXAMPLES:
sage:
sage:
sage:
[256,
from sage.rings.qqbar import tail_prec_seq
g = tail_prec_seq()
[next(g), next(g), next(g)]
512, 1024]
5.2 Universal cyclotomic field
The universal cyclotomic field is the smallest subfield of the complex field containing all roots of unity. It is also the
maximal Galois Abelian extension of the rational numbers.
The implementation simply wraps GAP Cyclotomic. As mentioned in their documentation: arithmetical operations
are quite expensive, so the use of internally represented cyclotomics is not recommended for doing arithmetic over
number fields, such as calculations with matrices of cyclotomics.
Note: There used to be a native Sage version of the universal cyclotomic field written by Christian Stump (see trac
ticket #8327). It was slower on most operations and it was decided to use a version based on libGAP instead (see
trac ticket #18152). One main difference in the design choices is that GAP stores dense vectors whereas the native
ones used Python dictionaries (storing only nonzero coefficients). Most operations are faster with libGAP except some
operation on very sparse elements. All details can be found in trac ticket #18152.
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REFERENCES:
EXAMPLES:
sage: UCF = UniversalCyclotomicField(); UCF
Universal Cyclotomic Field
To generate cyclotomic elements:
sage: UCF.gen(5)
E(5)
sage: UCF.gen(5,2)
E(5)^2
sage: E = UCF.gen
Equality and inequality checks:
sage: E(6,2) == E(6)^2 == E(3)
True
sage: E(6)^2 != E(3)
False
Addition and multiplication:
sage: E(2) * E(3)
-E(3)
sage: f = E(2) + E(3); f
2*E(3) + E(3)^2
Inverses:
sage: f^-1
1/3*E(3) + 2/3*E(3)^2
sage: f.inverse()
1/3*E(3) + 2/3*E(3)^2
sage: f * f.inverse()
1
Conjugation and Galois conjugates:
sage: f.conjugate()
E(3) + 2*E(3)^2
sage: f.galois_conjugates()
[2*E(3) + E(3)^2, E(3) + 2*E(3)^2]
sage: f.norm_of_galois_extension()
3
One can create matrices and polynomials:
sage: m = matrix(2,[E(3),1,1,E(4)]); m
[E(3)
1]
[
1 E(4)]
sage: m.parent()
Full MatrixSpace of 2 by 2 dense matrices over Universal Cyclotomic Field
sage: m**2
[
-E(3) E(12)^4 - E(12)^7 - E(12)^11]
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[E(12)^4 - E(12)^7 - E(12)^11
0]
sage: m.charpoly()
x^2 + (-E(12)^4 + E(12)^7 + E(12)^11)*x + E(12)^4 + E(12)^7 + E(12)^8
sage: m.echelon_form()
[1 0]
[0 1]
sage: m.pivots()
(0, 1)
sage: m.rank()
2
sage: R.<x> = PolynomialRing(UniversalCyclotomicField(), 'x')
sage: E(3) * x - 1
E(3)*x - 1
AUTHORS:
• Christian Stump (2013): initial Sage version (see trac ticket #8327)
• Vincent Delecroix (2015): complete rewriting using libgap (see trac ticket #18152)
sage.rings.universal_cyclotomic_field. E ( n, k=1)
Return the n -th root of unity as an element of the universal cyclotomic field.
EXAMPLES:
sage: E(3)
E(3)
sage: E(3) + E(5)
-E(15)^2 - 2*E(15)^8 - E(15)^11 - E(15)^13 - E(15)^14
class sage.rings.universal_cyclotomic_field. UCFtoQQbar ( UCF)
Bases: sage.categories.morphism.Morphism
Conversion to QQbar .
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: QQbar(UCF.gen(3))
-0.500000000000000? + 0.866025403784439?*I
sage: CC(UCF.gen(7,2) + UCF.gen(7,6))
0.400968867902419 + 0.193096429713794*I
sage: complex(E(7)+E(7,2))
(0.40096886790241915+1.7567593946498534j)
sage: complex(UCF.one()/2)
(0.5+0j)
class sage.rings.universal_cyclotomic_field. UniversalCyclotomicField ( names=None)
Bases:
sage.structure.unique_representation.UniqueRepresentation
,
sage.rings.ring.Field
The universal cyclotomic field.
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The universal cyclotomic field is the infinite algebraic extension of Q generated by the roots of unity. It is also
the maximal Abelian extension of Q in the sense that any Abelian Galois extension of Q is also a subfield of
the universal cyclotomic field.
Element
alias of UniversalCyclotomicFieldElement
algebraic_closure ( )
The algebraic closure.
EXAMPLES:
sage: UniversalCyclotomicField().algebraic_closure()
Algebraic Field
an_element ( )
Return an element.
EXAMPLES:
sage: UniversalCyclotomicField().an_element()
E(5) - 3*E(5)^2
characteristic ( )
Return the characteristic.
EXAMPLES:
sage: UniversalCyclotomicField().characteristic()
0
sage: parent(_)
Integer Ring
degree ( )
Return the degree of self as a field extension over the Rationals.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.degree()
+Infinity
gen ( n, k=1)
Return the standard primitive n -th root of unity.
If k is not None , return the k -th power of it.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(15)
E(15)
sage: UCF.gen(7,3)
E(7)^3
sage: UCF.gen(4,2)
-1
There is an alias zeta also available:
sage: UCF.zeta(6)
-E(3)^2
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is_exact ( )
Return True as this is an exact ring (i.e. not numerical).
EXAMPLES:
sage: UniversalCyclotomicField().is_exact()
True
is_finite ( )
Return True .
EXAMPLES:
sage: UniversalCyclotomicField().is_finite()
True
Todo
this method should be provided by the category.
one ( )
Return one.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.one()
1
sage: parent(_)
Universal Cyclotomic Field
some_elements ( )
Return a tuple of some elements in the universal cyclotomic field.
EXAMPLES:
sage: UniversalCyclotomicField().some_elements()
(0, 1, -1, E(3), E(7) - 2/3*E(7)^2)
sage: all(parent(x) is UniversalCyclotomicField() for x in _)
True
zero ( )
Return zero.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.zero()
0
sage: parent(_)
Universal Cyclotomic Field
zeta ( n, k=1)
Return the standard primitive n -th root of unity.
If k is not None , return the k -th power of it.
EXAMPLES:
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sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(15)
E(15)
sage: UCF.gen(7,3)
E(7)^3
sage: UCF.gen(4,2)
-1
There is an alias zeta also available:
sage: UCF.zeta(6)
-E(3)^2
class sage.rings.universal_cyclotomic_field. UniversalCyclotomicFieldElement ( parent,
obj)
Bases: sage.structure.element.FieldElement
INPUT:
•parent - a universal cyclotomic field
•obj - a libgap element (either an integer, a rational or a cyclotomic)
additive_order ( )
The additive order.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.zero().additive_order()
0
sage: UCF.one().additive_order()
+Infinity
sage: UCF.gen(3).additive_order()
+Infinity
conductor ( )
Return the conductor of self .
EXAMPLES:
sage: E(3).conductor()
3
sage: (E(5) + E(3)).conductor()
15
conjugate ( )
Return the complex conjugate.
EXAMPLES:
sage: (E(7) + 3*E(7,2) - 5 * E(7,3)).conjugate()
-5*E(7)^4 + 3*E(7)^5 + E(7)^6
denominator ( )
Return the denominator of this element.
EXAMPLES:
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sage: a = E(5) + 1/2*E(5,2) + 1/3*E(5,3)
sage: a
E(5) + 1/2*E(5)^2 + 1/3*E(5)^3
sage: a.denominator()
6
sage: parent(_)
Integer Ring
field_order ( *args, **kwds)
Deprecated: Use conductor() instead. See trac ticket #18152 for details.
galois_conjugates ( n=None)
Return the Galois conjugates of self .
INPUT:
•n – an optional integer. If provided, return the orbit of the Galois group of the n -th cyclotomic field
over Q. Note that n must be such that this element belongs to the n -th cyclotomic field (in other
words, it must be a multiple of the conductor).
EXAMPLES:
sage: E(6).galois_conjugates()
[-E(3)^2, -E(3)]
sage: E(6).galois_conjugates()
[-E(3)^2, -E(3)]
sage: (E(9,2) - E(9,4)).galois_conjugates()
[E(9)^2 - E(9)^4,
E(9)^2 + E(9)^4 + E(9)^5,
-E(9)^2 - E(9)^5 - E(9)^7,
-E(9)^2 - E(9)^4 - E(9)^7,
E(9)^4 + E(9)^5 + E(9)^7,
-E(9)^5 + E(9)^7]
sage: zeta = E(5)
sage: zeta.galois_conjugates(5)
[E(5), E(5)^2, E(5)^3, E(5)^4]
sage: zeta.galois_conjugates(10)
[E(5), E(5)^3, E(5)^2, E(5)^4]
sage: zeta.galois_conjugates(15)
[E(5), E(5)^2, E(5)^4, E(5)^2, E(5)^3, E(5), E(5)^3, E(5)^4]
sage: zeta.galois_conjugates(17)
Traceback (most recent call last):
...
ValueError: n = 17 must be a multiple of the conductor (5)
imag ( )
Return the imaginary part of this element.
EXAMPLES:
sage: E(3).imag()
-1/2*E(12)^7 + 1/2*E(12)^11
sage: E(5).imag()
1/2*E(20) - 1/2*E(20)^9
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sage: a = E(5) - 2*E(3)
sage: AA(a.imag()) == QQbar(a).imag()
True
imag_part ( )
Return the imaginary part of this element.
EXAMPLES:
sage: E(3).imag()
-1/2*E(12)^7 + 1/2*E(12)^11
sage: E(5).imag()
1/2*E(20) - 1/2*E(20)^9
sage: a = E(5) - 2*E(3)
sage: AA(a.imag()) == QQbar(a).imag()
True
inverse ( )
is_rational ( )
Test whether this element is a rational number.
EXAMPLES:
sage: E(3).is_rational()
False
sage: (E(3) + E(3,2)).is_rational()
True
is_real ( )
Test whether this element is real.
EXAMPLES:
sage: E(3).is_real()
False
sage: (E(3) + E(3,2)).is_real()
True
sage: a = E(3) - 2*E(7)
sage: a.real_part().is_real()
True
sage: a.imag_part().is_real()
True
minpoly ( var=’x’)
The minimal polynomial of self element over Q.
INPUT:
•var – (optional, default ‘x’) the name of the variable to use.
EXAMPLES:
sage: UCF.<E> = UniversalCyclotomicField()
sage: UCF(4).minpoly()
x - 4
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sage: UCF(4).minpoly(var='y')
y - 4
sage: E(3).minpoly()
x^2 + x + 1
sage: E(3).minpoly(var='y')
y^2 + y + 1
Todo
Polynomials with libgap currently does not implement a .sage() method (see trac ticket #18266). It
would be faster/safer to not use string to construct the polynomial.
multiplicative_order ( )
The multiplicative order.
EXAMPLES:
sage: E(5).multiplicative_order()
5
sage: (E(5) + E(12)).multiplicative_order()
+Infinity
sage: UniversalCyclotomicField().zero().multiplicative_order()
Traceback (most recent call last):
...
ValueError: libGAP: Error, argument must be nonzero
norm_of_galois_extension ( )
Return the norm as a Galois extension of Q, which is given by the product of all galois_conjugates.
EXAMPLES:
sage: E(3).norm_of_galois_extension()
1
sage: E(6).norm_of_galois_extension()
1
sage: (E(2) + E(3)).norm_of_galois_extension()
3
sage: parent(_)
Integer Ring
real ( )
Return the real part of this element.
EXAMPLES:
sage: E(3).real()
-1/2
sage: E(5).real()
1/2*E(5) + 1/2*E(5)^4
sage: a = E(5) - 2*E(3)
sage: AA(a.real()) == QQbar(a).real()
True
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real_part ( )
Return the real part of this element.
EXAMPLES:
sage: E(3).real()
-1/2
sage: E(5).real()
1/2*E(5) + 1/2*E(5)^4
sage: a = E(5) - 2*E(3)
sage: AA(a.real()) == QQbar(a).real()
True
to_cyclotomic_field ( R=None)
Return this element as an element of a cyclotomic field.
EXAMPLES:
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(3).to_cyclotomic_field()
zeta3
sage: UCF.gen(3,2).to_cyclotomic_field()
-zeta3 - 1
sage: CF = CyclotomicField(5)
sage: CF(E(5)) # indirect doctest
zeta5
sage: CF = CyclotomicField(7)
sage: CF(E(5)) # indirect doctest
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta5 into Cyclotomic Field of order 7 and
degree 6
sage: CF = CyclotomicField(10)
sage: CF(E(5)) # indirect doctest
zeta10^2
Matrices are correctly dealt with:
sage: M = Matrix(UCF,2,[E(3),E(4),E(5),E(6)]); M
[
E(3)
E(4)]
[
E(5) -E(3)^2]
sage: Matrix(CyclotomicField(60),M) # indirect doctest
[zeta60^10 - 1
zeta60^15]
[
zeta60^12
zeta60^10]
Using a non-standard embedding:
sage: CF = CyclotomicField(5,embedding=CC(exp(4*pi*i/5)))
sage: x = E(5)
sage: CC(x)
0.309016994374947 + 0.951056516295154*I
sage: CC(CF(x))
0.309016994374947 + 0.951056516295154*I
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Test that the bug reported in trac ticket #19912 has been fixed:
sage: a = 1+E(4); a
1 + E(4)
sage: a.to_cyclotomic_field()
zeta4 + 1
sage.rings.universal_cyclotomic_field. late_import ( )
This function avoids importing libgap on startup.
It is called once through the constructor of
UniversalCyclotomicField .
EXAMPLES:
sage: import sage.rings.universal_cyclotomic_field as ucf
sage: _ = UniversalCyclotomicField()
# indirect doctest
sage: ucf.libgap is None
# indirect doctest
False
See also:
Algebraic closures of finite fields
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CHAPTER
SIX
INDICES AND TABLES
• Index
• Module Index
• Search Page
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BIBLIOGRAPHY
[Doyle-Krumm] John R. Doyle and David Krumm, Computing algebraic numbers of bounded height, Arxiv
1111.4963 (2013).
[C] H. Cohen. A Course in Computational Algebraic Number Theory. Springer-Verlag, 1993.
[Cohen2000] Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol.
193, Springer-Verlag, New York, 2000.
[Martinet1980] Jacques Martinet, Petits discriminants des corps de nombres, Journ. Arithm. 1980, Cambridge Univ.
Press, 1982, 151–193.
[Takeuchi1999] Kisao Takeuchi, Totally real algebraic number fields of degree 9 with small discriminant, Saitama
Math. J. 17 (1999), 63–85 (2000).
[Voight2008] John Voight, Enumeration of totally real number fields of bounded root discriminant, Lect. Notes in
Comp. Sci. 5011 (2008).
[CE] Henry Cohn and Noam Elkies, New upper bounds on sphere packings I, Ann. Math. 157 (2003), 689–714.
[CS] J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, 3rd. ed., Grundlehren der Mathematischen
Wissenschaften, vol. 290, Springer-Verlag, New York, 1999.
[Bre97] T. Breuer “Integral bases for subfields of cyclotomic fields” AAECC 8, 279–289 (1997).
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PYTHON MODULE INDEX
r
sage.rings.number_field.bdd_height, 172
sage.rings.number_field.class_group, 252
sage.rings.number_field.galois_group, 166
sage.rings.number_field.maps, 187
sage.rings.number_field.morphism, 179
sage.rings.number_field.number_field, 1
sage.rings.number_field.number_field_base, 91
sage.rings.number_field.number_field_element, 117
sage.rings.number_field.number_field_element_quadratic, 152
sage.rings.number_field.number_field_ideal, 215
sage.rings.number_field.number_field_ideal_rel, 242
sage.rings.number_field.number_field_morphisms, 183
sage.rings.number_field.number_field_rel, 94
sage.rings.number_field.order, 195
sage.rings.number_field.small_primes_of_degree_one, 263
sage.rings.number_field.splitting_field, 161
sage.rings.number_field.structure, 192
sage.rings.number_field.totallyreal, 267
sage.rings.number_field.totallyreal_data, 274
sage.rings.number_field.totallyreal_phc, 277
sage.rings.number_field.totallyreal_rel, 270
sage.rings.number_field.unit_group, 257
sage.rings.qqbar, 279
sage.rings.universal_cyclotomic_field, 332
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Python Module Index
INDEX
A
abs() (sage.rings.number_field.number_field_element.NumberFieldElement method), 117
abs() (sage.rings.qqbar.ANDescr method), 289
abs() (sage.rings.qqbar.ANExtensionElement method), 291
abs() (sage.rings.qqbar.ANRational method), 295
abs_hom() (sage.rings.number_field.morphism.RelativeNumberFieldHomomorphism_from_abs method), 181
abs_non_arch() (sage.rings.number_field.number_field_element.NumberFieldElement method), 119
absolute_base_field() (sage.rings.number_field.number_field_rel.NumberField_relative method), 96
absolute_charpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 144
absolute_charpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 147
absolute_charpoly() (sage.rings.number_field.number_field_element.OrderElement_relative method), 150
absolute_degree() (sage.rings.number_field.number_field.NumberField_absolute method), 12
absolute_degree() (sage.rings.number_field.number_field.NumberField_generic method), 39
absolute_degree() (sage.rings.number_field.number_field_rel.NumberField_relative method), 96
absolute_degree() (sage.rings.number_field.order.Order method), 200
absolute_different() (sage.rings.number_field.number_field.NumberField_absolute method), 12
absolute_different() (sage.rings.number_field.number_field_rel.NumberField_relative method), 97
absolute_discriminant() (sage.rings.number_field.number_field.NumberField_absolute method), 12
absolute_discriminant() (sage.rings.number_field.number_field_rel.NumberField_relative method), 97
absolute_discriminant() (sage.rings.number_field.order.AbsoluteOrder method), 196
absolute_discriminant() (sage.rings.number_field.order.RelativeOrder method), 209
absolute_field() (sage.rings.number_field.number_field.NumberField_generic method), 39
absolute_field() (sage.rings.number_field.number_field_rel.NumberField_relative method), 97
absolute_generator() (sage.rings.number_field.number_field.NumberField_absolute method), 12
absolute_generator() (sage.rings.number_field.number_field_rel.NumberField_relative method), 98
absolute_ideal() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 243
absolute_minpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 145
absolute_minpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 147
absolute_minpoly() (sage.rings.number_field.number_field_element.OrderElement_relative method), 150
absolute_norm() (sage.rings.number_field.number_field_element.NumberFieldElement method), 119
absolute_norm() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 229
absolute_norm() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 243
absolute_order() (sage.rings.number_field.order.AbsoluteOrder method), 196
absolute_order() (sage.rings.number_field.order.RelativeOrder method), 210
absolute_order_from_module_generators() (in module sage.rings.number_field.order), 211
absolute_order_from_ring_generators() (in module sage.rings.number_field.order), 213
absolute_polynomial() (sage.rings.number_field.number_field.NumberField_absolute method), 12
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absolute_polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 98
absolute_polynomial_ntl() (sage.rings.number_field.number_field.NumberField_generic method), 39
absolute_polynomial_ntl() (sage.rings.number_field.number_field_rel.NumberField_relative method), 99
absolute_ramification_index() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 229
absolute_ramification_index()
(sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel
method), 244
absolute_vector_space() (sage.rings.number_field.number_field.NumberField_absolute method), 13
absolute_vector_space() (sage.rings.number_field.number_field_rel.NumberField_relative method), 99
AbsoluteFromRelative (class in sage.rings.number_field.structure), 192
AbsoluteOrder (class in sage.rings.number_field.order), 195
additive_order() (sage.rings.number_field.number_field_element.NumberFieldElement method), 119
additive_order() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 337
algebraic_closure() (sage.rings.number_field.number_field.NumberField_generic method), 39
algebraic_closure() (sage.rings.qqbar.AlgebraicField method), 301
algebraic_closure() (sage.rings.qqbar.AlgebraicRealField method), 322
algebraic_closure() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
AlgebraicField (class in sage.rings.qqbar), 301
AlgebraicField_common (class in sage.rings.qqbar), 304
AlgebraicGenerator (class in sage.rings.qqbar), 305
AlgebraicGeneratorRelation (class in sage.rings.qqbar), 308
AlgebraicNumber (class in sage.rings.qqbar), 308
AlgebraicNumber_base (class in sage.rings.qqbar), 310
AlgebraicPolynomialTracker (class in sage.rings.qqbar), 317
AlgebraicReal (class in sage.rings.qqbar), 318
AlgebraicRealField (class in sage.rings.qqbar), 322
alpha() (sage.rings.number_field.number_field_element.CoordinateFunction method), 117
ambient() (sage.rings.number_field.order.Order method), 200
ambient_field (sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldConversion attribute), 183
ambient_field (sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldMorphism attribute), 184
an_binop_element() (in module sage.rings.qqbar), 324
an_binop_expr() (in module sage.rings.qqbar), 325
an_binop_rational() (in module sage.rings.qqbar), 325
an_element() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
ANBinaryExpr (class in sage.rings.qqbar), 287
ANDescr (class in sage.rings.qqbar), 289
ANExtensionElement (class in sage.rings.qqbar), 290
angle() (sage.rings.qqbar.ANRational method), 295
ANRational (class in sage.rings.qqbar), 295
ANRoot (class in sage.rings.qqbar), 297
ANRootOfUnity (class in sage.rings.qqbar), 299
ANUnaryExpr (class in sage.rings.qqbar), 299
artin_symbol() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 169
artin_symbol() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 229
as_hom() (sage.rings.number_field.galois_group.GaloisGroupElement method), 167
as_number_field_element() (sage.rings.qqbar.AlgebraicNumber_base method), 311
automorphisms() (sage.rings.number_field.number_field.NumberField_absolute method), 13
automorphisms() (sage.rings.number_field.number_field_rel.NumberField_relative method), 99
B
bach_bound() (sage.rings.number_field.number_field_base.NumberField method), 91
350
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
base_field() (sage.rings.number_field.number_field.NumberField_absolute method), 14
base_field() (sage.rings.number_field.number_field_rel.NumberField_relative method), 100
base_ring() (sage.rings.number_field.number_field_rel.NumberField_relative method), 101
basis() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 230
basis() (sage.rings.number_field.order.AbsoluteOrder method), 196
basis() (sage.rings.number_field.order.Order method), 201
basis() (sage.rings.number_field.order.RelativeOrder method), 210
basis_to_module() (in module sage.rings.number_field.number_field_ideal), 240
bdd_height() (in module sage.rings.number_field.bdd_height), 172
bdd_height_iq() (in module sage.rings.number_field.bdd_height), 174
bdd_norm_pr_gens_iq() (in module sage.rings.number_field.bdd_height), 175
bdd_norm_pr_ideal_gens() (in module sage.rings.number_field.bdd_height), 176
C
cardinality() (sage.rings.number_field.morphism.NumberFieldHomset method), 180
ceil() (sage.rings.number_field.number_field_element.NumberFieldElement method), 120
ceil() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 152
ceil() (sage.rings.qqbar.AlgebraicReal method), 318
change_generator() (sage.rings.number_field.number_field.NumberField_generic method), 39
change_names() (sage.rings.number_field.number_field.NumberField_absolute method), 14
change_names() (sage.rings.number_field.number_field_rel.NumberField_relative method), 101
change_names() (sage.rings.number_field.order.AbsoluteOrder method), 197
characteristic() (sage.rings.number_field.number_field.NumberField_generic method), 40
characteristic() (sage.rings.qqbar.AlgebraicField_common method), 304
characteristic() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
charpoly() (sage.rings.number_field.number_field_element.NumberFieldElement method), 120
charpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 145
charpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 148
charpoly() (sage.rings.number_field.number_field_element.OrderElement_relative method), 150
charpoly() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 153
charpoly() (sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic method), 160
class_group() (sage.rings.number_field.number_field.NumberField_generic method), 40
class_group() (sage.rings.number_field.order.Order method), 201
class_number() (sage.rings.number_field.number_field.NumberField_generic method), 41
class_number() (sage.rings.number_field.number_field.NumberField_quadratic method), 83
class_number() (sage.rings.number_field.order.Order method), 201
ClassGroup (class in sage.rings.number_field.class_group), 252
clear_denominators() (in module sage.rings.qqbar), 326
closest() (in module sage.rings.number_field.number_field_morphisms), 185
coefficients_to_power_sums() (in module sage.rings.number_field.totallyreal_phc), 277
common_polynomial() (sage.rings.qqbar.AlgebraicField_common method), 304
completion() (sage.rings.number_field.number_field.NumberField_generic method), 42
completion() (sage.rings.qqbar.AlgebraicField method), 301
completion() (sage.rings.qqbar.AlgebraicRealField method), 322
complex_conjugation() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 169
complex_conjugation() (sage.rings.number_field.number_field.NumberField_generic method), 42
complex_embedding() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 30
complex_embedding() (sage.rings.number_field.number_field_element.NumberFieldElement method), 120
complex_embeddings() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 31
complex_embeddings() (sage.rings.number_field.number_field.NumberField_generic method), 42
Index
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complex_embeddings() (sage.rings.number_field.number_field_element.NumberFieldElement method), 121
complex_exact() (sage.rings.qqbar.AlgebraicNumber method), 308
complex_number() (sage.rings.qqbar.AlgebraicNumber method), 308
complex_roots() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 317
composite_fields() (sage.rings.number_field.number_field.NumberField_generic method), 43
composite_fields() (sage.rings.number_field.number_field_rel.NumberField_relative method), 102
conductor() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 337
conjugate() (sage.rings.number_field.number_field_element.NumberFieldElement method), 121
conjugate() (sage.rings.qqbar.AlgebraicGenerator method), 306
conjugate() (sage.rings.qqbar.AlgebraicNumber method), 309
conjugate() (sage.rings.qqbar.AlgebraicReal method), 319
conjugate() (sage.rings.qqbar.ANDescr method), 289
conjugate() (sage.rings.qqbar.ANExtensionElement method), 291
conjugate() (sage.rings.qqbar.ANRoot method), 297
conjugate() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 337
conjugate_expand() (in module sage.rings.qqbar), 326
conjugate_shrink() (in module sage.rings.qqbar), 326
construction() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 31
construction() (sage.rings.number_field.number_field.NumberField_generic method), 44
construction() (sage.rings.qqbar.AlgebraicField method), 302
continued_fraction()
(sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic
method), 153
continued_fraction_list() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic
method), 153
CoordinateFunction (class in sage.rings.number_field.number_field_element), 117
coordinates() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 230
coordinates() (sage.rings.number_field.order.Order method), 201
coordinates_in_terms_of_powers() (sage.rings.number_field.number_field_element.NumberFieldElement method),
122
create_embedding_from_approx() (in module sage.rings.number_field.number_field_morphisms), 185
create_key() (sage.rings.number_field.number_field.CyclotomicFieldFactory method), 4
create_key_and_extra_args() (sage.rings.number_field.number_field.NumberFieldFactory method), 9
create_object() (sage.rings.number_field.number_field.CyclotomicFieldFactory method), 4
create_object() (sage.rings.number_field.number_field.NumberFieldFactory method), 9
create_structure() (sage.rings.number_field.structure.AbsoluteFromRelative method), 193
create_structure() (sage.rings.number_field.structure.NameChange method), 193
create_structure() (sage.rings.number_field.structure.NumberFieldStructure method), 193
create_structure() (sage.rings.number_field.structure.RelativeFromAbsolute method), 193
create_structure() (sage.rings.number_field.structure.RelativeFromRelative method), 193
CyclotomicFieldEmbedding (class in sage.rings.number_field.number_field_morphisms), 183
CyclotomicFieldFactory (class in sage.rings.number_field.number_field), 2
CyclotomicFieldHomomorphism_im_gens (class in sage.rings.number_field.morphism), 179
CyclotomicFieldHomset (class in sage.rings.number_field.morphism), 179
D
decomposition_group() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 170
decomposition_group() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 231
default_base_hom() (sage.rings.number_field.morphism.RelativeNumberFieldHomset method), 182
default_interval_prec() (sage.rings.qqbar.AlgebraicField_common method), 305
defining_polynomial() (sage.rings.number_field.number_field.NumberField_generic method), 45
352
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
defining_polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 103
degree() (sage.rings.number_field.number_field.NumberField_generic method), 45
degree() (sage.rings.number_field.number_field_base.NumberField method), 92
degree() (sage.rings.number_field.number_field_rel.NumberField_relative method), 103
degree() (sage.rings.number_field.order.Order method), 202
degree() (sage.rings.qqbar.AlgebraicNumber_base method), 312
degree() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
denominator() (sage.rings.number_field.number_field_element.NumberFieldElement method), 122
denominator() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method),
153
denominator() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 215
denominator() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 337
denominator_ideal() (sage.rings.number_field.number_field_element.NumberFieldElement method), 123
descend_mod_power() (sage.rings.number_field.number_field_element.NumberFieldElement method), 123
different() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 32
different() (sage.rings.number_field.number_field.NumberField_generic method), 45
different() (sage.rings.number_field.number_field_rel.NumberField_relative method), 103
disc() (sage.rings.number_field.number_field.NumberField_generic method), 46
disc() (sage.rings.number_field.number_field_rel.NumberField_relative method), 103
discriminant() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 32
discriminant() (sage.rings.number_field.number_field.NumberField_generic method), 46
discriminant() (sage.rings.number_field.number_field.NumberField_quadratic method), 83
discriminant() (sage.rings.number_field.number_field_base.NumberField method), 92
discriminant() (sage.rings.number_field.number_field_rel.NumberField_relative method), 103
discriminant() (sage.rings.number_field.order.AbsoluteOrder method), 197
divides() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 216
do_polred() (in module sage.rings.qqbar), 327
E
E() (in module sage.rings.universal_cyclotomic_field), 334
each_is_integral() (in module sage.rings.number_field.order), 214
easy_is_irreducible_py() (in module sage.rings.number_field.totallyreal_data), 274
EisensteinIntegers() (in module sage.rings.number_field.order), 198
Element (sage.rings.number_field.class_group.ClassGroup attribute), 253
Element (sage.rings.number_field.class_group.SClassGroup attribute), 256
Element (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField attribute), 335
element_1_mod() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 216
element_1_mod() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 244
elements_of_bounded_height() (sage.rings.number_field.number_field.NumberField_absolute method), 14
elements_of_norm() (sage.rings.number_field.number_field.NumberField_generic method), 46
EmbeddedNumberFieldConversion (class in sage.rings.number_field.number_field_morphisms), 183
EmbeddedNumberFieldMorphism (class in sage.rings.number_field.number_field_morphisms), 183
embeddings() (sage.rings.number_field.number_field.NumberField_absolute method), 16
embeddings() (sage.rings.number_field.number_field_rel.NumberField_relative method), 104
enumerate_totallyreal_fields_all() (in module sage.rings.number_field.totallyreal_rel), 271
enumerate_totallyreal_fields_prim() (in module sage.rings.number_field.totallyreal), 268
enumerate_totallyreal_fields_rel() (in module sage.rings.number_field.totallyreal_rel), 272
EquationOrder() (in module sage.rings.number_field.order), 199
euler_phi() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 216
exactify() (sage.rings.qqbar.AlgebraicNumber_base method), 312
Index
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Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
exactify() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 317
exactify() (sage.rings.qqbar.ANBinaryExpr method), 287
exactify() (sage.rings.qqbar.ANExtensionElement method), 291
exactify() (sage.rings.qqbar.ANRational method), 295
exactify() (sage.rings.qqbar.ANRoot method), 297
exactify() (sage.rings.qqbar.ANUnaryExpr method), 299
exp() (sage.rings.number_field.unit_group.UnitGroup method), 260
extension() (sage.rings.number_field.number_field.NumberField_generic method), 47
F
factor() (sage.rings.number_field.number_field.NumberField_generic method), 47
factor() (sage.rings.number_field.number_field_element.NumberFieldElement method), 124
factor() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 217
factor() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 244
factors() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 318
field() (sage.rings.qqbar.AlgebraicGenerator method), 306
field_element_value() (sage.rings.qqbar.ANExtensionElement method), 291
field_order() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 338
find_zero_result() (in module sage.rings.qqbar), 327
fixed_field() (sage.rings.number_field.galois_group.GaloisGroup_subgroup method), 167
floor() (sage.rings.number_field.number_field_element.NumberFieldElement method), 124
floor() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 154
floor() (sage.rings.qqbar.AlgebraicReal method), 319
fraction_field() (sage.rings.number_field.order.Order method), 202
fractional_ideal() (sage.rings.number_field.number_field.NumberField_generic method), 48
fractional_ideal() (sage.rings.number_field.order.Order method), 202
FractionalIdealClass (class in sage.rings.number_field.class_group), 253
free_module() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 231
free_module() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 245
free_module() (sage.rings.number_field.order.Order method), 203
fundamental_units() (sage.rings.number_field.unit_group.UnitGroup method), 260
G
galois_closure() (sage.rings.number_field.number_field.NumberField_absolute method), 17
galois_closure() (sage.rings.number_field.number_field_rel.NumberField_relative method), 104
galois_conjugate()
(sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic
method), 154
galois_conjugates() (sage.rings.number_field.number_field_element.NumberFieldElement method), 125
galois_conjugates() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 338
galois_group() (sage.rings.number_field.number_field.NumberField_generic method), 49
galois_group() (sage.rings.number_field.number_field_rel.NumberField_relative method), 104
GaloisGroup (in module sage.rings.number_field.galois_group), 166
GaloisGroup_subgroup (class in sage.rings.number_field.galois_group), 167
GaloisGroup_v1 (class in sage.rings.number_field.galois_group), 168
GaloisGroup_v2 (class in sage.rings.number_field.galois_group), 169
GaloisGroupElement (class in sage.rings.number_field.galois_group), 167
GaussianIntegers() (in module sage.rings.number_field.order), 199
gcd() (sage.rings.number_field.number_field_element.NumberFieldElement method), 126
gen() (sage.rings.number_field.number_field.NumberField_generic method), 51
gen() (sage.rings.number_field.number_field_rel.NumberField_relative method), 105
354
Index
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gen() (sage.rings.number_field.order.Order method), 203
gen() (sage.rings.qqbar.AlgebraicField method), 302
gen() (sage.rings.qqbar.AlgebraicRealField method), 323
gen() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
gen_embedding() (sage.rings.number_field.number_field.NumberField_generic method), 51
gen_image() (sage.rings.number_field.number_field_morphisms.NumberFieldEmbedding method), 185
generator() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 318
generator() (sage.rings.qqbar.ANExtensionElement method), 291
generator() (sage.rings.qqbar.ANRational method), 295
gens() (sage.rings.number_field.class_group.FractionalIdealClass method), 254
gens() (sage.rings.number_field.number_field_rel.NumberField_relative method), 105
gens() (sage.rings.number_field.order.Order method), 203
gens() (sage.rings.qqbar.AlgebraicField method), 302
gens() (sage.rings.qqbar.AlgebraicRealField method), 323
gens_ideals() (sage.rings.number_field.class_group.ClassGroup method), 253
gens_reduced() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 232
gens_reduced() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 245
gens_two() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 233
get_AA_golden_ratio() (in module sage.rings.qqbar), 328
global_height() (sage.rings.number_field.number_field_element.NumberFieldElement method), 127
global_height_arch() (sage.rings.number_field.number_field_element.NumberFieldElement method), 127
global_height_non_arch() (sage.rings.number_field.number_field_element.NumberFieldElement method), 128
group() (sage.rings.number_field.galois_group.GaloisGroup_v1 method), 168
H
handle_sage_input() (sage.rings.qqbar.ANBinaryExpr method), 287
handle_sage_input() (sage.rings.qqbar.ANExtensionElement method), 292
handle_sage_input() (sage.rings.qqbar.ANRational method), 295
handle_sage_input() (sage.rings.qqbar.ANRoot method), 298
handle_sage_input() (sage.rings.qqbar.ANUnaryExpr method), 300
hermite_constant() (in module sage.rings.number_field.totallyreal_data), 275
hilbert_class_field() (sage.rings.number_field.number_field.NumberField_quadratic method), 84
hilbert_class_field_defining_polynomial() (sage.rings.number_field.number_field.NumberField_quadratic method),
84
hilbert_class_polynomial() (sage.rings.number_field.number_field.NumberField_quadratic method), 85
hilbert_conductor() (sage.rings.number_field.number_field.NumberField_absolute method), 18
hilbert_symbol() (sage.rings.number_field.number_field.NumberField_absolute method), 18
I
ideal() (sage.rings.number_field.class_group.FractionalIdealClass method), 254
ideal() (sage.rings.number_field.number_field.NumberField_generic method), 51
ideal() (sage.rings.number_field.order.Order method), 204
ideal_below() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 245
ideal_class_log() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 233
idealcoprime() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 217
ideallog() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 218
ideals_of_bdd_norm() (sage.rings.number_field.number_field.NumberField_generic method), 51
idealstar() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 219
im_gens() (sage.rings.number_field.morphism.RelativeNumberFieldHomomorphism_from_abs method), 182
imag() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 154
Index
355
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
imag() (sage.rings.qqbar.AlgebraicNumber method), 309
imag() (sage.rings.qqbar.AlgebraicReal method), 319
imag() (sage.rings.qqbar.ANDescr method), 289
imag() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 338
imag_part() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 339
incr() (sage.rings.number_field.totallyreal_rel.tr_data_rel method), 274
increment() (sage.rings.number_field.totallyreal_data.tr_data method), 276
index_in() (sage.rings.number_field.order.AbsoluteOrder method), 197
index_in() (sage.rings.number_field.order.RelativeOrder method), 210
inertia_group() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 170
inertia_group() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 234
int_has_small_square_divisor() (in module sage.rings.number_field.totallyreal_data), 275
integer_points_in_polytope() (in module sage.rings.number_field.bdd_height), 176
integral_basis() (sage.rings.number_field.number_field.NumberField_generic method), 52
integral_basis() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 234
integral_basis() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 247
integral_closure() (sage.rings.number_field.order.Order method), 204
integral_elements_in_box() (in module sage.rings.number_field.totallyreal_rel), 273
integral_split() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 234
integral_split() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 247
intersection() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 234
intersection() (sage.rings.number_field.order.AbsoluteOrder method), 198
interval() (sage.rings.qqbar.AlgebraicNumber_base method), 312
interval_diameter() (sage.rings.qqbar.AlgebraicNumber_base method), 313
interval_exact() (sage.rings.qqbar.AlgebraicNumber method), 309
interval_exact() (sage.rings.qqbar.AlgebraicReal method), 319
interval_fast() (sage.rings.qqbar.AlgebraicNumber_base method), 313
inverse() (sage.rings.number_field.class_group.FractionalIdealClass method), 254
inverse() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 339
inverse_mod() (sage.rings.number_field.number_field_element.NumberFieldElement method), 128
inverse_mod() (sage.rings.number_field.number_field_element.OrderElement_absolute method), 149
inverse_mod() (sage.rings.number_field.number_field_element.OrderElement_relative method), 151
inverse_mod() (sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic method), 160
invert() (sage.rings.qqbar.ANDescr method), 289
invert() (sage.rings.qqbar.ANExtensionElement method), 292
invert() (sage.rings.qqbar.ANRational method), 296
invertible_residues() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 219
invertible_residues_mod() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 220
is_absolute() (sage.rings.number_field.number_field.NumberField_absolute method), 22
is_absolute() (sage.rings.number_field.number_field.NumberField_generic method), 54
is_absolute() (sage.rings.number_field.number_field_base.NumberField method), 92
is_absolute() (sage.rings.number_field.number_field_rel.NumberField_relative method), 106
is_AbsoluteNumberField() (in module sage.rings.number_field.number_field), 87
is_AlgebraicField() (in module sage.rings.qqbar), 328
is_AlgebraicField_common() (in module sage.rings.qqbar), 328
is_AlgebraicNumber() (in module sage.rings.qqbar), 328
is_AlgebraicReal() (in module sage.rings.qqbar), 328
is_AlgebraicRealField() (in module sage.rings.qqbar), 329
is_CM() (sage.rings.number_field.number_field.NumberField_generic method), 52
is_CM_extension() (sage.rings.number_field.number_field_rel.NumberField_relative method), 105
356
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
is_complex() (sage.rings.qqbar.AlgebraicGenerator method), 306
is_complex() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 318
is_complex() (sage.rings.qqbar.ANBinaryExpr method), 289
is_complex() (sage.rings.qqbar.ANExtensionElement method), 293
is_complex() (sage.rings.qqbar.ANRational method), 296
is_complex() (sage.rings.qqbar.ANRoot method), 299
is_complex() (sage.rings.qqbar.ANUnaryExpr method), 301
is_coprime() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 222
is_CyclotomicField() (in module sage.rings.number_field.number_field), 87
is_exact() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 335
is_field() (sage.rings.number_field.number_field.NumberField_generic method), 54
is_field() (sage.rings.number_field.order.Order method), 205
is_finite() (sage.rings.number_field.number_field_base.NumberField method), 92
is_finite() (sage.rings.qqbar.AlgebraicField_common method), 305
is_finite() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 336
is_free() (sage.rings.number_field.number_field_rel.NumberField_relative method), 106
is_fundamental_discriminant() (in module sage.rings.number_field.number_field), 88
is_galois() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 170
is_galois() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 32
is_galois() (sage.rings.number_field.number_field.NumberField_generic method), 54
is_galois() (sage.rings.number_field.number_field.NumberField_quadratic method), 85
is_galois() (sage.rings.number_field.number_field_rel.NumberField_relative method), 106
is_galois_absolute() (sage.rings.number_field.number_field_rel.NumberField_relative method), 106
is_galois_relative() (sage.rings.number_field.number_field_rel.NumberField_relative method), 107
is_injective() (sage.rings.number_field.maps.NumberFieldIsomorphism method), 191
is_integer() (sage.rings.number_field.number_field_element.NumberFieldElement method), 129
is_integer() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 155
is_integer() (sage.rings.qqbar.AlgebraicNumber_base method), 314
is_integral() (sage.rings.number_field.number_field_element.NumberFieldElement method), 129
is_integral() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method),
155
is_integral() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 235
is_integral() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 247
is_integrally_closed() (sage.rings.number_field.order.Order method), 205
is_isomorphic() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 32
is_isomorphic() (sage.rings.number_field.number_field.NumberField_generic method), 54
is_isomorphic_relative() (sage.rings.number_field.number_field_rel.NumberField_relative method), 107
is_maximal() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 223
is_maximal() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 235
is_maximal() (sage.rings.number_field.order.Order method), 205
is_noetherian() (sage.rings.number_field.order.Order method), 205
is_norm() (sage.rings.number_field.number_field_element.NumberFieldElement method), 130
is_nth_power() (sage.rings.number_field.number_field_element.NumberFieldElement method), 131
is_NumberField() (in module sage.rings.number_field.number_field_base), 94
is_NumberFieldElement() (in module sage.rings.number_field.number_field_element), 152
is_NumberFieldFractionalIdeal() (in module sage.rings.number_field.number_field_ideal), 240
is_NumberFieldFractionalIdeal_rel() (in module sage.rings.number_field.number_field_ideal_rel), 251
is_NumberFieldHomsetCodomain() (in module sage.rings.number_field.number_field), 87
is_NumberFieldIdeal() (in module sage.rings.number_field.number_field_ideal), 241
is_NumberFieldOrder() (in module sage.rings.number_field.order), 214
Index
357
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
is_one() (sage.rings.number_field.number_field_element.NumberFieldElement method), 131
is_one() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 155
is_prime() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 235
is_prime() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 248
is_principal() (sage.rings.number_field.class_group.FractionalIdealClass method), 255
is_principal() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 236
is_principal() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 248
is_QuadraticField() (in module sage.rings.number_field.number_field), 88
is_rational() (sage.rings.number_field.number_field_element.NumberFieldElement method), 131
is_rational() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method),
156
is_rational() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 339
is_real() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 339
is_real_positive() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 145
is_relative() (sage.rings.number_field.number_field.NumberField_generic method), 54
is_RelativeNumberField() (in module sage.rings.number_field.number_field_rel), 116
is_S_integral() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 221
is_S_unit() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 221
is_simple() (sage.rings.qqbar.ANDescr method), 290
is_simple() (sage.rings.qqbar.ANExtensionElement method), 293
is_simple() (sage.rings.qqbar.ANRational method), 296
is_square() (sage.rings.number_field.number_field_element.NumberFieldElement method), 132
is_square() (sage.rings.qqbar.AlgebraicNumber_base method), 314
is_suborder() (sage.rings.number_field.order.Order method), 206
is_suborder() (sage.rings.number_field.order.RelativeOrder method), 211
is_surjective() (sage.rings.number_field.maps.NumberFieldIsomorphism method), 192
is_totally_imaginary() (sage.rings.number_field.number_field.NumberField_generic method), 55
is_totally_positive() (sage.rings.number_field.number_field_element.NumberFieldElement method), 132
is_totally_real() (sage.rings.number_field.number_field.NumberField_generic method), 55
is_trivial() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 223
is_trivial() (sage.rings.qqbar.AlgebraicGenerator method), 306
is_unit() (sage.rings.number_field.number_field_element.NumberFieldElement method), 132
is_zero() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 236
is_zero() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 248
isolating_interval() (in module sage.rings.qqbar), 329
K
key() (sage.rings.number_field.splitting_field.SplittingData method), 162
krull_dimension() (sage.rings.number_field.order.Order method), 206
L
lagrange_degree_3() (in module sage.rings.number_field.totallyreal_data), 275
late_import() (in module sage.rings.universal_cyclotomic_field), 342
latex_variable_name() (sage.rings.number_field.number_field.NumberField_generic method), 55
lift() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 146
lift() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 148
lift_to_base() (sage.rings.number_field.number_field_rel.NumberField_relative method), 108
LiftMap (class in sage.rings.number_field.number_field_ideal), 215
list() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
list() (sage.rings.number_field.morphism.CyclotomicFieldHomset method), 179
358
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
list() (sage.rings.number_field.morphism.NumberFieldHomset method), 180
list() (sage.rings.number_field.morphism.RelativeNumberFieldHomset method), 182
list() (sage.rings.number_field.number_field_element.NumberFieldElement method), 133
list() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 146
list() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 148
local_height() (sage.rings.number_field.number_field_element.NumberFieldElement method), 133
local_height_arch() (sage.rings.number_field.number_field_element.NumberFieldElement method), 134
log() (sage.rings.number_field.unit_group.UnitGroup method), 261
M
MapAbsoluteToRelativeNumberField (class in sage.rings.number_field.maps), 187
MapNumberFieldToVectorSpace (class in sage.rings.number_field.maps), 187
MapRelativeNumberFieldToRelativeVectorSpace (class in sage.rings.number_field.maps), 187
MapRelativeNumberFieldToVectorSpace (class in sage.rings.number_field.maps), 188
MapRelativeToAbsoluteNumberField (class in sage.rings.number_field.maps), 188
MapRelativeVectorSpaceToRelativeNumberField (class in sage.rings.number_field.maps), 189
MapVectorSpaceToNumberField (class in sage.rings.number_field.maps), 190
MapVectorSpaceToRelativeNumberField (class in sage.rings.number_field.maps), 191
matching_root() (in module sage.rings.number_field.number_field_morphisms), 186
matrix() (sage.rings.number_field.number_field_element.NumberFieldElement method), 134
maximal_order() (sage.rings.number_field.number_field.NumberField_absolute method), 22
maximal_order() (sage.rings.number_field.number_field_base.NumberField method), 93
maximal_order() (sage.rings.number_field.number_field_rel.NumberField_relative method), 108
maximal_totally_real_subfield() (sage.rings.number_field.number_field.NumberField_generic method), 55
minkowski_bound() (sage.rings.number_field.number_field_base.NumberField method), 93
Minkowski_embedding() (sage.rings.number_field.number_field.NumberField_absolute method), 11
minpoly() (sage.rings.number_field.number_field_element.NumberFieldElement method), 136
minpoly() (sage.rings.number_field.number_field_element.NumberFieldElement_absolute method), 146
minpoly() (sage.rings.number_field.number_field_element.OrderElement_relative method), 151
minpoly() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 156
minpoly() (sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic method), 160
minpoly() (sage.rings.qqbar.AlgebraicNumber_base method), 314
minpoly() (sage.rings.qqbar.ANExtensionElement method), 293
minpoly() (sage.rings.qqbar.ANRational method), 296
minpoly() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 339
module() (sage.rings.number_field.order.AbsoluteOrder method), 198
multiplicative_order() (sage.rings.number_field.number_field_element.NumberFieldElement method), 136
multiplicative_order() (sage.rings.qqbar.AlgebraicNumber method), 309
multiplicative_order() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 340
N
NameChange (class in sage.rings.number_field.structure), 193
NameChangeMap (class in sage.rings.number_field.maps), 191
narrow_class_group() (sage.rings.number_field.number_field.NumberField_generic method), 57
neg() (sage.rings.qqbar.ANDescr method), 290
neg() (sage.rings.qqbar.ANExtensionElement method), 294
neg() (sage.rings.qqbar.ANRational method), 296
next() (sage.rings.number_field.small_primes_of_degree_one.Small_primes_of_degree_one_iter method), 265
next_split_prime() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 33
ngens() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
Index
359
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
ngens() (sage.rings.number_field.number_field.NumberField_generic method), 57
ngens() (sage.rings.number_field.number_field_rel.NumberField_relative method), 109
ngens() (sage.rings.number_field.order.Order method), 206
ngens() (sage.rings.qqbar.AlgebraicField method), 302
ngens() (sage.rings.qqbar.AlgebraicRealField method), 323
norm() (sage.rings.number_field.number_field_element.NumberFieldElement method), 137
norm() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 156
norm() (sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic method), 161
norm() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 236
norm() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 248
norm() (sage.rings.qqbar.AlgebraicNumber method), 310
norm() (sage.rings.qqbar.ANDescr method), 290
norm() (sage.rings.qqbar.ANExtensionElement method), 294
norm_of_galois_extension() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 340
nth_root() (sage.rings.number_field.number_field_element.NumberFieldElement method), 138
nth_root() (sage.rings.qqbar.AlgebraicNumber_base method), 314
number_field() (sage.rings.number_field.class_group.ClassGroup method), 253
number_field() (sage.rings.number_field.galois_group.GaloisGroup_v1 method), 168
number_field() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
number_field() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 236
number_field() (sage.rings.number_field.order.Order method), 207
number_field() (sage.rings.number_field.unit_group.UnitGroup method), 261
number_field_elements_from_algebraics() (in module sage.rings.qqbar), 329
number_of_roots_of_unity() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 33
number_of_roots_of_unity() (sage.rings.number_field.number_field.NumberField_generic method), 57
number_of_roots_of_unity() (sage.rings.number_field.number_field_rel.NumberField_relative method), 109
NumberField (class in sage.rings.number_field.number_field_base), 91
NumberField() (in module sage.rings.number_field.number_field), 4
NumberField_absolute (class in sage.rings.number_field.number_field), 11
NumberField_absolute_v1() (in module sage.rings.number_field.number_field), 29
NumberField_cyclotomic (class in sage.rings.number_field.number_field), 29
NumberField_cyclotomic_v1() (in module sage.rings.number_field.number_field), 35
NumberField_extension_v1() (in module sage.rings.number_field.number_field_rel), 95
NumberField_generic (class in sage.rings.number_field.number_field), 35
NumberField_generic_v1() (in module sage.rings.number_field.number_field), 82
NumberField_quadratic (class in sage.rings.number_field.number_field), 82
NumberField_quadratic_v1() (in module sage.rings.number_field.number_field), 85
NumberField_relative (class in sage.rings.number_field.number_field_rel), 96
NumberField_relative_v1() (in module sage.rings.number_field.number_field_rel), 116
NumberFieldElement (class in sage.rings.number_field.number_field_element), 117
NumberFieldElement_absolute (class in sage.rings.number_field.number_field_element), 144
NumberFieldElement_quadratic (class in sage.rings.number_field.number_field_element_quadratic), 152
NumberFieldElement_relative (class in sage.rings.number_field.number_field_element), 147
NumberFieldEmbedding (class in sage.rings.number_field.number_field_morphisms), 184
NumberFieldFactory (class in sage.rings.number_field.number_field), 8
NumberFieldFractionalIdeal (class in sage.rings.number_field.number_field_ideal), 215
NumberFieldFractionalIdeal_rel (class in sage.rings.number_field.number_field_ideal_rel), 242
NumberFieldHomomorphism_im_gens (class in sage.rings.number_field.morphism), 179
NumberFieldHomset (class in sage.rings.number_field.morphism), 180
NumberFieldIdeal (class in sage.rings.number_field.number_field_ideal), 229
360
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
NumberFieldIsomorphism (class in sage.rings.number_field.maps), 191
NumberFieldStructure (class in sage.rings.number_field.structure), 193
NumberFieldTower() (in module sage.rings.number_field.number_field), 9
numerator() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic
157
numerator() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 223
numerator_ideal() (sage.rings.number_field.number_field_element.NumberFieldElement method), 138
method),
O
odlyzko_bound_totallyreal() (in module sage.rings.number_field.totallyreal), 269
OK() (sage.rings.number_field.number_field_base.NumberField method), 91
one() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 336
optimized_representation() (sage.rings.number_field.number_field.NumberField_absolute method), 22
optimized_subfields() (sage.rings.number_field.number_field.NumberField_absolute method), 23
ord() (sage.rings.number_field.number_field_element.NumberFieldElement method), 138
Order (class in sage.rings.number_field.order), 200
order() (sage.rings.number_field.galois_group.GaloisGroup_v1 method), 168
order() (sage.rings.number_field.morphism.NumberFieldHomset method), 181
order() (sage.rings.number_field.number_field.NumberField_absolute method), 24
order() (sage.rings.number_field.number_field.NumberField_generic method), 57
order() (sage.rings.number_field.number_field_rel.NumberField_relative method), 109
order() (sage.rings.qqbar.AlgebraicField_common method), 305
OrderElement_absolute (class in sage.rings.number_field.number_field_element), 149
OrderElement_quadratic (class in sage.rings.number_field.number_field_element_quadratic), 160
OrderElement_relative (class in sage.rings.number_field.number_field_element), 150
P
pari_absolute_base_polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 110
pari_bnf() (sage.rings.number_field.number_field.NumberField_generic method), 57
pari_field() (sage.rings.qqbar.AlgebraicGenerator method), 306
pari_hnf() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 236
pari_nf() (sage.rings.number_field.number_field.NumberField_generic method), 58
pari_polynomial() (sage.rings.number_field.number_field.NumberField_generic method), 59
pari_prime() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 237
pari_relative_polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 110
pari_rhnf() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 248
pari_rnf() (sage.rings.number_field.number_field_rel.NumberField_relative method), 111
pari_rnfnorm_data() (sage.rings.number_field.number_field.NumberField_generic method), 60
pari_zk() (sage.rings.number_field.number_field.NumberField_generic method), 60
parts() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 157
places() (sage.rings.number_field.number_field.NumberField_absolute method), 25
places() (sage.rings.number_field.number_field_rel.NumberField_relative method), 111
poldegree() (sage.rings.number_field.splitting_field.SplittingData method), 162
poly() (sage.rings.qqbar.AlgebraicPolynomialTracker method), 318
polynomial() (sage.rings.number_field.number_field.NumberField_generic method), 61
polynomial() (sage.rings.number_field.number_field_element.NumberFieldElement method), 139
polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 111
polynomial_ntl() (sage.rings.number_field.number_field.NumberField_generic method), 61
polynomial_quotient_ring() (sage.rings.number_field.number_field.NumberField_generic method), 61
polynomial_ring() (sage.rings.number_field.number_field.NumberField_generic method), 61
Index
361
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
polynomial_root() (sage.rings.qqbar.AlgebraicField method), 302
polynomial_root() (sage.rings.qqbar.AlgebraicRealField method), 323
power_basis() (sage.rings.number_field.number_field.NumberField_generic method), 61
prec_seq() (in module sage.rings.qqbar), 331
preimage() (sage.rings.number_field.morphism.NumberFieldHomomorphism_im_gens method), 179
prime_above() (sage.rings.number_field.number_field.NumberField_generic method), 62
prime_factors() (sage.rings.number_field.number_field.NumberField_generic method), 63
prime_factors() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 223
prime_to_idealM_part() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 224
prime_to_S_part() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 224
primes() (sage.rings.number_field.unit_group.UnitGroup method), 262
primes_above() (sage.rings.number_field.number_field.NumberField_generic method), 63
primes_of_bounded_norm() (sage.rings.number_field.number_field.NumberField_generic method), 65
primes_of_bounded_norm_iter() (sage.rings.number_field.number_field.NumberField_generic method), 65
primes_of_degree_one_iter() (sage.rings.number_field.number_field.NumberField_generic method), 66
primes_of_degree_one_list() (sage.rings.number_field.number_field.NumberField_generic method), 66
primitive_element() (sage.rings.number_field.number_field.NumberField_generic method), 67
primitive_root_of_unity() (sage.rings.number_field.number_field.NumberField_generic method), 67
printa() (sage.rings.number_field.totallyreal_data.tr_data method), 276
proof_flag() (in module sage.rings.number_field.number_field), 88
put_natural_embedding_first() (in module sage.rings.number_field.number_field), 89
Q
Q_to_quadratic_field_element (class in sage.rings.number_field.number_field_element_quadratic), 161
QuadraticField() (in module sage.rings.number_field.number_field), 85
quotient_char_p() (in module sage.rings.number_field.number_field_ideal), 241
QuotientMap (class in sage.rings.number_field.number_field_ideal), 240
R
radical_expression() (sage.rings.qqbar.AlgebraicNumber_base method), 315
ramification_breaks() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
ramification_degree() (sage.rings.number_field.galois_group.GaloisGroupElement method), 167
ramification_group() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
ramification_group() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 237
ramification_index() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 225
ramification_index() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 249
random_element() (sage.rings.number_field.number_field.NumberField_generic method), 68
random_element() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 237
random_element() (sage.rings.number_field.order.Order method), 207
random_element() (sage.rings.qqbar.AlgebraicField method), 303
rank() (sage.rings.number_field.order.Order method), 208
rank() (sage.rings.number_field.unit_group.UnitGroup method), 262
rational_argument() (sage.rings.qqbar.AlgebraicNumber method), 310
rational_argument() (sage.rings.qqbar.ANExtensionElement method), 294
rational_argument() (sage.rings.qqbar.ANRational method), 297
rational_exact_root() (in module sage.rings.qqbar), 332
ray_class_number() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 225
real() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 158
real() (sage.rings.qqbar.AlgebraicNumber method), 310
real() (sage.rings.qqbar.AlgebraicReal method), 320
362
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
real() (sage.rings.qqbar.ANDescr method), 290
real() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 340
real_embeddings() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 33
real_embeddings() (sage.rings.number_field.number_field.NumberField_generic method), 68
real_exact() (sage.rings.qqbar.AlgebraicReal method), 320
real_number() (sage.rings.qqbar.AlgebraicReal method), 321
real_part() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 340
real_places() (sage.rings.number_field.number_field.NumberField_absolute method), 26
reduce() (sage.rings.number_field.class_group.FractionalIdealClass method), 255
reduce() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 225
reduce_equiv() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 238
reduced_basis() (sage.rings.number_field.number_field.NumberField_generic method), 69
reduced_gram_matrix() (sage.rings.number_field.number_field.NumberField_generic method), 70
refine_embedding() (in module sage.rings.number_field.number_field), 89
refine_interval() (sage.rings.qqbar.ANRoot method), 299
regulator() (sage.rings.number_field.number_field.NumberField_generic method), 71
relative_degree() (sage.rings.number_field.number_field.NumberField_absolute method), 26
relative_degree() (sage.rings.number_field.number_field_rel.NumberField_relative method), 112
relative_different() (sage.rings.number_field.number_field.NumberField_absolute method), 26
relative_different() (sage.rings.number_field.number_field_rel.NumberField_relative method), 112
relative_discriminant() (sage.rings.number_field.number_field.NumberField_absolute method), 26
relative_discriminant() (sage.rings.number_field.number_field_rel.NumberField_relative method), 112
relative_norm() (sage.rings.number_field.number_field_element.NumberFieldElement method), 139
relative_norm() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 238
relative_norm() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 249
relative_order_from_ring_generators() (in module sage.rings.number_field.order), 214
relative_polynomial() (sage.rings.number_field.number_field.NumberField_absolute method), 26
relative_polynomial() (sage.rings.number_field.number_field_rel.NumberField_relative method), 112
relative_ramification_index() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 238
relative_ramification_index()
(sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel
method), 249
relative_vector_space() (sage.rings.number_field.number_field.NumberField_absolute method), 26
relative_vector_space() (sage.rings.number_field.number_field_rel.NumberField_relative method), 112
RelativeFromAbsolute (class in sage.rings.number_field.structure), 193
RelativeFromRelative (class in sage.rings.number_field.structure), 193
RelativeNumberFieldHomomorphism_from_abs (class in sage.rings.number_field.morphism), 181
RelativeNumberFieldHomset (class in sage.rings.number_field.morphism), 182
RelativeOrder (class in sage.rings.number_field.order), 209
relativize() (sage.rings.number_field.number_field.NumberField_absolute method), 27
relativize() (sage.rings.number_field.number_field_rel.NumberField_relative method), 113
representative_prime() (sage.rings.number_field.class_group.FractionalIdealClass method), 255
residue_class_degree() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 226
residue_class_degree() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method),
250
residue_field() (sage.rings.number_field.number_field.NumberField_generic method), 71
residue_field() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 226
residue_field() (sage.rings.number_field.order.Order method), 208
residue_symbol() (sage.rings.number_field.number_field_element.NumberFieldElement method), 139
residue_symbol() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 238
residues() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 228
Index
363
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
residues() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 250
ring_generators() (sage.rings.number_field.order.Order method), 209
ring_of_integers() (sage.rings.number_field.number_field_base.NumberField method), 94
root_as_algebraic() (sage.rings.qqbar.AlgebraicGenerator method), 307
root_from_approx() (in module sage.rings.number_field.number_field_morphisms), 186
roots_of_unity() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 33
roots_of_unity() (sage.rings.number_field.number_field.NumberField_generic method), 72
roots_of_unity() (sage.rings.number_field.number_field_rel.NumberField_relative method), 114
roots_of_unity() (sage.rings.number_field.unit_group.UnitGroup method), 262
round() (sage.rings.number_field.number_field_element.NumberFieldElement method), 141
round() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 158
round() (sage.rings.qqbar.AlgebraicReal method), 321
S
S() (sage.rings.number_field.class_group.SClassGroup method), 256
S_class_group() (sage.rings.number_field.number_field.NumberField_generic method), 36
S_ideal_class_log() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 229
S_unit_group() (sage.rings.number_field.number_field.NumberField_generic method), 36
S_units() (sage.rings.number_field.number_field.NumberField_generic method), 38
sage.rings.number_field.bdd_height (module), 172
sage.rings.number_field.class_group (module), 252
sage.rings.number_field.galois_group (module), 166
sage.rings.number_field.maps (module), 187
sage.rings.number_field.morphism (module), 179
sage.rings.number_field.number_field (module), 1
sage.rings.number_field.number_field_base (module), 91
sage.rings.number_field.number_field_element (module), 117
sage.rings.number_field.number_field_element_quadratic (module), 152
sage.rings.number_field.number_field_ideal (module), 215
sage.rings.number_field.number_field_ideal_rel (module), 242
sage.rings.number_field.number_field_morphisms (module), 183
sage.rings.number_field.number_field_rel (module), 94
sage.rings.number_field.order (module), 195
sage.rings.number_field.small_primes_of_degree_one (module), 263
sage.rings.number_field.splitting_field (module), 161
sage.rings.number_field.structure (module), 192
sage.rings.number_field.totallyreal (module), 267
sage.rings.number_field.totallyreal_data (module), 274
sage.rings.number_field.totallyreal_phc (module), 277
sage.rings.number_field.totallyreal_rel (module), 270
sage.rings.number_field.unit_group (module), 257
sage.rings.qqbar (module), 279
sage.rings.universal_cyclotomic_field (module), 332
scale() (sage.rings.qqbar.ANRational method), 297
SClassGroup (class in sage.rings.number_field.class_group), 256
section() (sage.rings.number_field.number_field_morphisms.EmbeddedNumberFieldMorphism method), 184
selmer_group() (sage.rings.number_field.number_field.NumberField_generic method), 72
selmer_group_iterator() (sage.rings.number_field.number_field.NumberField_generic method), 73
SFractionalIdealClass (class in sage.rings.number_field.class_group), 256
short_prec_seq() (in module sage.rings.qqbar), 332
364
Index
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
sign() (sage.rings.number_field.number_field_element.NumberFieldElement method), 141
sign() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 158
sign() (sage.rings.qqbar.AlgebraicReal method), 321
signature() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 34
signature() (sage.rings.number_field.number_field.NumberField_generic method), 74
signature() (sage.rings.number_field.number_field_base.NumberField method), 94
simplify() (sage.rings.qqbar.AlgebraicNumber_base method), 316
simplify() (sage.rings.qqbar.ANExtensionElement method), 294
Small_primes_of_degree_one_iter (class in sage.rings.number_field.small_primes_of_degree_one), 265
small_residue() (sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal method), 228
smallest_integer() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 239
smallest_integer() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 250
solve_CRT() (sage.rings.number_field.number_field.NumberField_generic method), 74
some_elements() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 336
specified_complex_embedding() (sage.rings.number_field.number_field.NumberField_generic method), 74
splitting_field() (in module sage.rings.number_field.splitting_field), 162
splitting_field() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 171
SplittingData (class in sage.rings.number_field.splitting_field), 161
SplittingFieldAbort, 162
sqrt() (sage.rings.number_field.number_field_element.NumberFieldElement method), 142
sqrt() (sage.rings.qqbar.AlgebraicNumber_base method), 316
structure() (sage.rings.number_field.number_field.NumberField_generic method), 75
subfield() (sage.rings.number_field.number_field.NumberField_generic method), 76
subfields() (sage.rings.number_field.number_field.NumberField_absolute method), 28
subfields() (sage.rings.number_field.number_field_rel.NumberField_relative method), 114
subgroup() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 172
super_poly() (sage.rings.qqbar.AlgebraicGenerator method), 307
support() (sage.rings.number_field.number_field_element.NumberFieldElement method), 142
T
t1 (in module sage.rings.qqbar), 332
t2 (in module sage.rings.qqbar), 332
tail_prec_seq() (in module sage.rings.qqbar), 332
timestr() (in module sage.rings.number_field.totallyreal), 270
to_cyclotomic_field() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicFieldElement method), 341
torsion_generator() (sage.rings.number_field.unit_group.UnitGroup method), 262
tr_data (class in sage.rings.number_field.totallyreal_data), 276
tr_data_rel (class in sage.rings.number_field.totallyreal_rel), 274
trace() (sage.rings.number_field.number_field_element.NumberFieldElement method), 143
trace() (sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic method), 159
trace() (sage.rings.number_field.number_field_element_quadratic.OrderElement_quadratic method), 161
trace_dual_basis() (sage.rings.number_field.number_field.NumberField_generic method), 77
trace_pairing() (sage.rings.number_field.number_field.NumberField_generic method), 77
trunc() (sage.rings.qqbar.AlgebraicReal method), 322
U
UCFtoQQbar (class in sage.rings.universal_cyclotomic_field), 334
uniformizer() (sage.rings.number_field.number_field.NumberField_generic method), 78
uniformizer() (sage.rings.number_field.number_field_rel.NumberField_relative method), 115
union() (sage.rings.qqbar.AlgebraicGenerator method), 307
Index
365
Sage Reference Manual: Algebraic Numbers and Number Fields, Release 7.6
unit_group() (sage.rings.number_field.number_field.NumberField_generic method), 79
UnitGroup (class in sage.rings.number_field.unit_group), 260
units() (sage.rings.number_field.number_field.NumberField_generic method), 80
UniversalCyclotomicField (class in sage.rings.universal_cyclotomic_field), 334
UniversalCyclotomicFieldElement (class in sage.rings.universal_cyclotomic_field), 337
unrank() (sage.rings.number_field.galois_group.GaloisGroup_v2 method), 172
V
valuation() (sage.rings.number_field.number_field_element.NumberFieldElement method), 143
valuation() (sage.rings.number_field.number_field_element.NumberFieldElement_relative method), 149
valuation() (sage.rings.number_field.number_field_ideal.NumberFieldIdeal method), 240
valuation() (sage.rings.number_field.number_field_ideal_rel.NumberFieldFractionalIdeal_rel method), 251
vector() (sage.rings.number_field.number_field_element.NumberFieldElement method), 144
vector_space() (sage.rings.number_field.number_field.NumberField_absolute method), 29
vector_space() (sage.rings.number_field.number_field_rel.NumberField_relative method), 116
W
weed_fields() (in module sage.rings.number_field.totallyreal), 270
Z
Z_to_quadratic_field_element (class in sage.rings.number_field.number_field_element_quadratic), 161
zero() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 336
zeta() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 34
zeta() (sage.rings.number_field.number_field.NumberField_generic method), 80
zeta() (sage.rings.number_field.order.Order method), 209
zeta() (sage.rings.number_field.unit_group.UnitGroup method), 262
zeta() (sage.rings.qqbar.AlgebraicField method), 304
zeta() (sage.rings.qqbar.AlgebraicRealField method), 324
zeta() (sage.rings.universal_cyclotomic_field.UniversalCyclotomicField method), 336
zeta_coefficients() (sage.rings.number_field.number_field.NumberField_generic method), 81
zeta_function() (sage.rings.number_field.number_field.NumberField_generic method), 81
zeta_order() (sage.rings.number_field.number_field.NumberField_cyclotomic method), 35
zeta_order() (sage.rings.number_field.number_field.NumberField_generic method), 82
zeta_order() (sage.rings.number_field.unit_group.UnitGroup method), 263
366
Index